Ill  f! 

!        I     :       !« 


BERKELEY 

LIBRARY 


or 


WASHINGTON    OBSERVATIONS    FOR    1870.— APPENDIX    III. 

. 


ON    THE    RIGHT    ASCENSIONS 


EQUATORIAL  FUNDAMENTAL  STARS 


AND   THE   CORRECTIONS   NKCIWSAKY   TO   REUUCK   THE 


RIGHT    ASCENSIONS    OF    DIFFERENT    CATALOGUES 


MEAN      HOMOGENEOUS      SYSTEM 


BY 
SIMON      NEWCOMB, 

PROFESSOR   OF    MATHEMATICS,    UNITED   STATES   NAVY. 


PREPARED  AT  THE  U.  S.  NAVAL  OBSERVATORY 

1!Y   ORDER   OF 

REAR-ADMIRAL   H.    F.   SANDS,    U.    S.    N., 

SUPERINTENDENT. 


WASHINGTON: 

GOVERNMENT    PRINTING    OFFICE. 
I  S  7  2  . 


TABLE    OF   CONTENTS. 


Pape. 

i  I.  Introduction 5 

S  2.  Determination  of  a   system  of  corrections  to  the  relative  right  ascensions  of  ISessel's  Tabuliv  Kegiomnnlana.',  pre 
sumed  to  be  free  from  periodic  errors 0 

$3.  Discussion  of  the  probable  error  of  these  determinations ....  17 

$4.  Systematic  discordances   among  the  catalogues  employed 21 

$  5.  Determination  of  the  common  correction  to  the  first  set  of  positions  on  account  of  equinox 21 

$  6.  Corrections  for  the  terms  multiplied  by  the  second  and  third  powers  of  the  time 34 

$  7.  Corrections  to  different  catalogues  to  reduce  them  to  the  preceding  system 39 

J  8.  Derivation  of  definitive  right  ascensions 47 

Table  of  right  ascensions  from  1750  to  1900 53 

Corrections  for  orbital  motion  of  Sirius  and  Procyon 69 

Hill's  formulae  for  the  secular  variation  of  the  annual  motion  of  the  stars 73 


ON    THE    RIGHT    ASCENSIONS 


EQUATORIAL    FUNDAMENTAL    STARS 


THE  CORRECTIONS  NECESSARY  TO  REDUCE  THE  RIGHT  ASCENSIONS  OF 

DIFFERENT     STAR-CATALOGUES    TO    A    MEAN 

HOMOGENEOUS     SYSTEM. 


INTRODUCTION. 

Within  the  past  few  years  very  important  additions  have  been  made  to  the  mate 
rial  at  our  disposal  for  an  accurate  determination  of  the  positions  and  proper  motions 
of  the  fundamental  stars.  These  comprise  Dr.  Amvers'  reduction  of  Bradley's  observa 
tions,  the  Pulkowa  catalogue  of  1845,  an(l  the  continuation  of  the  Greenwich  and 
Washington  observations.  It  is  not  too  much  to  say  that  this  additional  material  alone 
would  suffice  to  furnish  results  far  more  accurate  than  any  that  could  be  obtained  from 
the  older  data. 

The  object  of  the  present  work  is  to  do  for  the  right  ascensions  of  the  equatorial 
and  zodiacal  stars,  on  which  the  reductions  of  lunar  and  planetary  observations  depend, 
what  has  been  done  by  Dr.  Auwers  for  the  declinations,  namely,  to  furnish  the  data 
necessary  to  reduce  the  principal  original  catalogues  of  stars  to  a  homogeneous  system 
by  freeing  them  of  their  systematic  differences.  I  think  this  can  best  be  done  by  com 
paring  the  catalogue-positions  of  the  fundamental  stars  with  a  uniform  set  of  standard 
positions  as  free  as  possible  from  systematic  error,  and  to  consider  any  systematic  dis 
cordance  between  the  standard  and  the  catalogue  position  as  due  to  error  of  the  latter, 
and  correct  it  accordingly.  For  the  reason  already  mentioned,  it  will  not  be  admissible 
to  take  any  existing  catalogue  as  the  standard.  The  first  step  of  our  work  is,  therefore, 
the  preparation  of  a  system  of  right  ascensions  of  fundamental  stars  which  shall  be  as 
free  as  possible  from  the  suspicion  of  systematic  error. 

In  the  formation  of  such  a  catalogue  I  have  been  guided  by  certain  considerations 
respecting  the  probable  presence  of  systematic  errors  in  the  reduced  right  ascensions 
of  many  modern  catalogues.  It  has  long  been  the  custom,  in  the  reduction  of  right- 
ascension  observations,  to  deduce  a  correction  of  the  clock  from  adopted  right  ascen 
sions  of  stars,  and  to  consider  the  mean  correction  thus  deduced  as  applicable  to  all  the 
stars  observed,  the  clock-stai'S  included.  On  this  system  it  is  clear  that  if  the  assumed 
positions  of  all  the  clock-stars  used  on  any  one  night  are  affected  by  a  common  error, 


6  RIGHT   ASCENSIONS   OE   FUNDAMENTAL   STARS. 

that  same  error  will  be  carried  into  the  concluded  positions.  Thus  any  periodic  error 
depending-  on  the  right  ascension  will  tend  to  perpetuate  itself.  In  a  former  paper  I 
showed  that  if  the  clock-stars  observed  on  each  night  are  distributed  uniformly  over  a 
period  of  six  hours,  and  if  the  assumed  positions  are  affected  with  the  error— 

a  sin  11.  A.  -f- 1  cos  K.  A.  -f-  a1  sin  2  11.  A.  -f-  ?/  cos  2  R.  A. 
the  concluded  positions  will  be  affected  with  the  error— 

82  («  sin  R  A.  +  &  cos  R  A.)  +  -*,  <y  ,Sm  2  R  A.  +  V  cos  2  R  A.) 

Hence  the  error  of  single  period  will  be  cut  down  very  slowly  by  successive  revisions 
of  the  catalogue,  each  revision  reducing  it  by  less  than  one-fifth.  On  the  other  hand, 
the  error  of  double  period  Avill  rapidly  disappear. 

The  introduction  of  these  periodic  errors  is  very  easy,  even  where  the  greatest 
care  is  taken  in  the  reductions,  owing  to  the  diurnal  variation  of  the  conditions  to 
which  the  instrument,  the  clock,  and  the  observer  are  subjected.  If  the  pointing"  of 
the  instrument,  the  rate  of  the  clock,  or  the  personal  equation  of  the  observer  vary 
during  the  twenty-four  hours,  the  right  ascensions  deduced  will  be  affected  by  a  cor 
responding  error.  It  is  true  that  if  the  variation  continues  uniform  during  an  entire 
year,  and  if  the  observations  are  uniformly  distributed  throughout  the  year,  this  error 
will  be  eliminated  from  a  year's  work.  But  neither  of  these  conditions  is  completely 
fulfilled,  the  diurnal  change  of  temperature  being  greater  in  summer  than  in  winter, 
while  the  annual  period  in  the  length  of  the  night  and  the  degree  of  cloudiness  interferes 
Avith  the  continuity  of  the  system  of  observation. 

§2. 

DETERMINATION  OE  A  SYSTEM  OF  CORRECTIONS  TO  THE  RELATIVE  RIGHT 
ASCENSIONS  OF  BESSEL'S  TABULAE  REGIOMONTANvE,  PRESUMED  TO  BE 
FREE  FROM  PERIODIC  ERRORS. 

For  the  reasons  already  mentioned,  it  seems  best  to  deduce  the  periodic  error  of 
the  standard  catalogue  solely  from  determinations  which  are  entirely  independent  of 
any  previous  hypothesis  respecting  the  right  ascensions  of  clock-stars,  and  from  those 
in  which  there  is  reason  to  believe  that  the  periodic  error  has  been  entirely  eliminated. 
I  have  selected  the  following  twelve  original  catalogues  as  certainly  or  probably  fulfill 
ing  one  of  these  conditions : 

1.  Ameers'  Bradley,  1755. — Dr.  Auwers  has   very  kindly  placed  his  definitive 
results  for  relative  right  ascensions,  which  are  as  yet  unpublished,  at  my  disposal. 

2.  Piazzi,  1800. — " Praecipuarum  stellarum  inerrantium  positiones  media?  ineunte 
saeculo  XIX.    Panormi,  1814." 

3.  Maskclync,  1805. — Catalogue  found  in  Maskelyne's  Astronomical  Observations 
made  at  the  Royal  Observatory,  Greenwich,  from  1799  to  1810,  p.  112  of  part  2. 

4.  Ameers'  Cacciatore. — "Reduction  der  Beobachtungen  der  Fundamentalsterne 
am  Passageninstrument  der  Sternwarte  zu  Palermo  in  den  Jahren  1803  his  l&°5i  un(l 
Bestimmung  der  mittleren  Rectascensionen  fur  1805.     Von  Arthur  Auwers.     Publica 
tion  der  astronomischen  Gesellschaft,  V.     Leipzig,  1866." 


DETERMINATION    OF   A    SYSTEM   OF   CORRECTIONS.  7 

5.  Bcssd  I. — "Bestimmung  dor  geraden  Aufsteigungen  der  36  Maskelyneschen 
Fundamental-Sterne  fur  1815,  auf  Kb'nigsberger  Beobachtungen  gegriindet."    Berlin 
Memoirs  for  iSiS-'ig,  p.  19  of  part  2. 

6.  Besscl  II. — "  Neue  Untersuchungen   iibor  die  geraden  Aufsteigungen  der  36 
Fundamental-Sterne."     Berlin  Memoirs  for  1825,  p.  23  of  part  2. 

7.  Stntcc. — Catalogue  found  in  "Positiones  media1,"  p.  xxxxi. 

s.  Brinklcy. — Catalogue  found  in  the  Astronomische  Nachrichten,  Band  IV,  S.  102. 

9.  Argelander.     Corrections  yireii  ly  Besscl  in  Astronomische  NacJiricJitcn,  No.  5 1 5. — 
These  differ  from  the  positions  in  Argelander's   "DLX.   Stellartun  fixarum  positiones 
media1"  only  in  the  addition  of  the  third  decimal  of  seconds. 

10.  Ptilkoica,  1845. — "Observations  do  Poulkova,  publiees  par  Otto  Struve,  Direc- 
teur  de  1'observatoire  central  Nicolas.     Saint-Petersbourg,  1869."    Vol.  I,  p.  120. 

ir.  Greenwich,  1864. — "New  seven-year  catalogue  of  2760  stars  deduced  from 
observations  extending  from  1861  to  1867  at  the  Royal  Observatory,  Greenwich,  and 
reduced  to  the  epoch  1864."  (Appendix  II  to  the  Greenwich  observations  for  1868.) 

1 2.  WasJiiiiyton,  1870. — "Positions  of  fundamental  stars  deduced  from  observations 
made  at  the  United  States  Naval  Observatory  between  the  years  1862  and  1867." 
(Washington  observations  for  1867,  Appendix  III.) 

On  this  selection  I  remark  that,  I  find  no  positive  evidence  that,  the  catalogues  of 
Piazzi  and  Brinkley  are  really  independent  of  any  previous  determination,  and  that  1 
have  judged  them  to  be  so  because  the  modern  system  of  reduction  had  not,  then  come 
into  general  use.  Argelander,  Greenwich,  and  Washington  are  not  rigorously  inde 
pendent,  but  they  arc  included,  because,  in  the  case  of  each,  there  is  reason  to  believe 
that  the  systematic  errors  have  been  more  or  less  completely  eliminated.  The  follow 
ing  are  the  reasons  in  the  case  of  each  catalogue: 

An  examination  of  Argelander's  observations  shows  that  they  were  generally 
widely  distributed  throughout  the  twenty-four  hours,  his  habit  being  to  observe  both 
morning  and  evening.  From  such  a  day's  work  the  error  of  single  period  will  be 
almost  entirely  eliminated. 

The  Memoirs  of  the  Royal  Astronomical  Society,  vol.  34,  contain  a  paper  by  Mr. 
Stone,  in  which  the  stars  y  Pegasi,  Pollux,  Spica,  and  a.  Aquila?  are  compared,  with  the 
express  purpose  of  detecting  any  inequality  depending  on  the  right  ascension.  The 
resiilt  is  that  the  relative  positions  of  these  stars,  as  given  in  the  seven-year  catalogue 
for  1860,  require  the  following  corrections: 

/  Pegasi              Pollux,  -.001    or  — .014 

Pollux    —      Spica,  +.029  or  -(-.015 

Spica        -  a.  Aquihv,  -f-  .008  or  —  .005 

a  Aquihc  — y  Pegasi,  +-OI8  or  +-004 

The  first  column  shows  the  differences  independently  deduced ;  the  second  their 
values  when  each  is  diminished  by  a  constant  so  as  to  make  their  sum  zero. 

The  smallness  of  this  result  indicates  that  the  periodic  error,  if  it  exists  at  all,  must 
be  very  small,  and  it  ought  to  be  smaller  still  in  the  catalogue  of  1864,  which  includes 
still  another  revision. 


8  RIGHT  ASCENSIONS  OF   FUNDAMENTAL  STARS. 

In  the  case  of  Washington  ;i  special  effort  was  made  to  determine  the  periodic 
error  of  Dr.  Gould's  right  ascensions;  which  was  of  the  form — 

a  sin  R.  A.  +  b  cos  R  A. 

the  details  of  which  are  given  in  the  catalogue  cited.  The  determination  was  entirely 
satisfactory  except  in  one  point — a  large  diurnal  inequality  showed  itself  during  the 
second  year.  This  inequality  had  to  be  determined  on  the  supposition  that  it  remained 
constant  throughout  the  year,  an  hypothesis  which,  though  probable,  was  not  certain. 

Having  selected  our  data,  the  next  question  will  be  the  mode  of  combining.  It 
was  my  original  intention  to  compare  each  catalogue  with  Dr.  Gould's  standard,  and 
thus  obtain  a  correction  to  the  latter  of  the  form — 

a  sin  a  -\-l)  cos  a-\-c 

A  series  of  values  of  a,  b,  and  c  being  thus  found  for  the  epochs  of  the  various 
catalogues,  the  true  values  of  these  quantities  should  increase  uniformly  with  the  time. 
We  should  then  determine  a0,  a',  10,  V,  coj  c',  from  the  equations — 

a0  +  ft't  =  n 
1>0  +  Vt  -  I 

c0  +  c't  =  c 

of  which  we  should  .have  as  many  sets  as  there  were  catalogues  compared.  Having 
found  the  most  probable  values  of  a0,  a',  etc.,  the  residuals  left  after  substituting  them 
in  the  equations  given  by  each  catalogue  would  show  the  systematic  error  of  that  cat 
alogue. 

But  it  has  been  deemed  best  to  begin  by  determining  each  star  independently  of 
any  catalogue  whatever,  and  of  any  assumed  proper  motion.  This  is  done  most  easily 
and  elegantly  by  assigning  the  same  weight  to  all  the  stars  in  any  one  catalogue  ;  this 
weight  being  determined  not  by  the  general  accuracy  of  the  catalogue,  but  solely  by 
its  probable  freedom  from  systematic  errors.  It  will  then  be  unnecessary  to  apply  any 
reduction  on  account  of  eqitinox  in  this  preliminary  determination,  because,  if  all  tlie 
right  ascensions  of  any  one  catalogue  be  changed  by  a  constant,  that  change  will  affect 
all  the  concluded  positions  by  the  same  amount,  and  therefore  will  not  change  the  rel 
ative  positions  at  all. 

From  the  nature  of  the  case,  it  is  clearly  impossible  to  make  any  numerical  calcu 
lation  of  the  relative  weights  to  be  assigned  to  the  different  determinations.  This  is 
necessarily  a  matter  'of  individual  judgment.  Without  going  into  the  consideration  of 
each  particular  case,  I-  may  remark  that  the  following  weights  assigned  were  founded 
on  considerations  respecting  the  probable  freedom  of  the  instruments  from  sources  of 
systematic  error  and  the  care  with  which  systematic  error  was  avoided  in  the  observa 
tions  and  reductions,  probable  accidental  errors  not  being  taken  into  account,  and  that 
they  were  decided  on  before  any  comparison  of  results.  They  have,  therefore,  no 
necessary  analogy  with  the  weights  to  be  assigned  in  a  definitive  discussion,  after  sys 
tematic  error  has  been  allowed  for. 


DETERMINATION   OF   A   SYSTEM   OF   CORRECTIONS. 

Weights  assigned  to  observations. 


No. 

Catalogue. 

Year. 

Weight. 

I 

2 

Auwers'  Bradley    .    . 
I'iazzi     

1755 
1800 

« 
i 

3 

iSos 

4 

5 

Auwers'  Cacciotorc   . 
Bessel  I    ... 

1805 
181=; 

I 

6 

7 

Bessel  II  
Struve   .    . 

1823' 
iS"i 

3 

•7 

8 

Brinkley  

i82a 

9 
10 

Argclander  
Pulkowa   

1828 
i8j? 

2 

6 

ii 

1864 

2 

12 

Washington  

l87O 

1 

The  dates  here  given  to  the  catalogues  are  generally  supposed  to  be  about  the 
means  of  the  times  of  observation  on  which  they  depend.  The  principal  exceptions 
are  Brinkley  and  Washington.  The  mean  date  of  the  former  could  not  be  inferred 
with  certainty,  but  must  be  earlier  than  1824.  The  date  assigned  for  the  latter  is, 
through  inadvertence,  that  to  which  the  catalogue  was  reduced,  which  is  five  years 
later  than  the  actual  mean  of  the  times  of  observation.  The  reduction  to  the  epoch 
was,  however,  made  with  Dr.  Gould's  proper  motions,  which  arc  so  near  the  truth  that 
few  or  none  of  the  results  are  altered  the  thousandth  of  a  second  by  this  error.  I  have, 
therefore,  not  corrected  it, 

The  results  are  all  exhibited  in  the  form  of  corrections  to  the  mean  positions  and 
annual  variations  of  the  Tabula;  Regiomontanre  for  the  sake  of  convenience,  and  the 
principal  steps  of  the  computation  are  exhibited  in  the  following  tables.  The  first 
column  in  each  table  shows  the  correction  to  the  right  ascension  of  the  Tabula;  Regio- 
montanse  given  by  the  catalogue.  Following  one  star  through  the  several  catalogues, 
we  have  a  series  of  corrections  applicable  at  the  various  epochs.  The  true  correction 
being  of  the  form  x  -\-  y  t,  each  catalogue  result  gives  an  equation  of  condition  between 
x  and  y.  As  an  epoch  near  the  mean  leads  to  the  most  convenient  computations,  I 
have  chosen  the  year  1820  as  the  origin  of  time.  Let  us  now  represent  by— 


the  corrections  given  by  the  several  catalogues,  the  subscript  numbers  being  the  same 
as  those  of  the  catalogues  to  which  they  correspond  in  the  preceding  list.  Taking  the 
century  as  the  unit  of  time,  the  equations  of  condition  will  then  be  — 

./•  —  0.65  //  =  (\ 

./'  —  O.2O1J  —  r2 
x  —  o.  1  5  y  —  r  , 


etc.,    etc.,    etc. 


2 — F  s 


I0  RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS.     • 

The  solution  of  these  equations  by  least  squares,  using  the  weights  in  the  preceding 
list,  will  give  the  following  values  of  x  and  y: 

-\-  0.04  c.2  -  o.io  f2 

+    0.04    f;,  -   0.08     C3 

+  0.04  c4  -  0.08  c4 

+  0.04  r5  -  0.03  c, 

+  o.  1 1  c6  -  0.04  ct; 

+  o.  1 1  c,  —  0.03  c7 

+  0.03  ce  —o.o  i  c8 

+  0.08  r,,  +  o.o  i  c, 

+  0.19  c,0  +0.40  r10 

+  0.06  rn  +0.27  cn 

No  account  is  here  taken  of  the  discordant  equinoxes  of  the  various  catalogues, 
because,  all  the  stars  in  the  same  catalogue  having  the  same  weight,  a  change  of 
equinox  will  not  change  the  relative  positions  of  the  stars. 

In  the  selection  of  stars,  attention  has  been  at  first  confined  to  the  Maskelyne 
fundamental  stars  between  the  limits  of  declination  -f  40°  and  —  30°.  Several  stars 
have  to  be  omitted,  namely,  Procyon  and  Sirius,  on  account  of  the  irregularity  of 
their  proper  motions  which  has  been  so  exhaustively  investigated  by  Auwers  that  no 
further  discussion  of  their  positions  is  yet  necessary ;  Castor,  on  account  of  its  duplic 
ity,  and  the  confusion  of  the  observations  upon  it,  which  render  it  unfit  for  use  as  a 
standard  star ;  /?  Virginis,  a.'  Librae,  and  a.'  Capricorni,  because  they  have  of  late 
dropped  out  of  use. 

Under  each  catalogue,  in  the  following  table,  we  have,  in  column  c,  the  correction 
to  the  Tabulae  Regiomontanae  given  by  the  catalogue,  from  which  the  values  of  x  and 
y  are  deduced  by  the  preceding  formulae. 

The  value  of  the  correction  x  -f  t  y  is  next  computed  for  the  epoch  of  each  cata 
logue,  and  subtracted  from  the  correction  given  by  observation.  The  excess  of  the 
catalogue  result  thus  obtained  is  given  in  the  second  column  of  the  table.  To  correct 
the  residuals  for  discordance  of  equinoxes,  the  mean  value  of  the  numbers  in  this 
column  is  taken,  and  subtracted  from  the  individual  numbers.  The  remainders  repre 
sent  the  discordances  between  the  catalogue  and  the  correction  as  finally  deduced. 


DETERMINATION   OF  A   SYSTEM   OF   CORRECTIONS. 


I  I 


Corrections  to  rig/it  ascensions  given  by  standard  catalogues. 


Star. 

Bradley,  1755 

Piaxzi,  1800. 

f\ 

c' 

c" 

fl 

<V 

<V 

y     Pegasi  .... 

+     65 

0 

-     31 

54 

-  65 

4-     18 

a     Arietis  .... 

-     56 

+    40 

+       9 

-   lOS 

-  144 

-     61 

«     Ccti. 

4-      1$ 

4-      6.1 

4-        T1 

76 

g(3 

<7 

~      j^ 

i         ^4 

^        J  J 

7° 

J 

a     Taiiri     .... 

—       12 

+       51 

+      20 

-    194 

-   172 

89 

/?    Orionis. 

+      40 

+     33 

4-         2 

54 

-     30 

+     47 

i)    Tauri     .... 

+    152 

+     31 

O 

—     53 

-     TOO 

17 

a     Orionis. 

+      30 

+      22 

9 

7. 

13 

4-     70 

,3    Geminorum     . 

—    102 

+      41 

+      10 

-   173 

-  133 

-     50 

a     Hydrx  .... 

-    139 

4-     23 

-       3 

-   118 

-     So 

+       3 

«     Lconis  .... 

—   106 

+     43 

+       12 

-   170 

—    121 

38 

/?    Lconis  .... 

-     76 

+     34 

+       3     ! 

-     85 

-     68 

+     19 

«     Virginis 

42 

4-     2S 

3 

-     S6 

54 

+     29 

a     Bootis  .... 

-     36 

+     39 

4-       8 

-   164 

-   M3 

-     60 

a-    Libra:    .... 

+  190 

+     23 

-       8 

4-      30 

-     33 

+     50 

a     Corona} 

-     66 

+     37 

4-       6 

-    240 

—  208 

-  125 

a     Serpontis    . 

+    122 

+     35 

+       4 

-      56 

-     86 

-       3 

11     Scorpii  .... 

+       84 

+       4 

-     =7 

4-    122 

+     98 

4-   181 

«     Hcrculis     . 

16 

+     40 

+       9     ' 

-    225 

—    202 

-   119 

a    Ophiuchi    . 

-   162 

+     44 

+     13 

-    If)  I 

-  n6 

-     33 

a     Lyrne     .... 

—  •'30 

+      23 

-       3 

-    1  =  7 

—   108 

-     25 

y     Aquilrc. 

+   163 

+     29 

—         2 

-     35 

-     85 

-       2 

a     Aquilic. 

12 

+     18 

13 

-     5° 

37 

4-     46 

/}    Aquilx. 

+    MI 

+     26 

-       5 

-     f>7 

-    -89 

6 

a3  Capricorn!. 

'7 

+      7 

24 

—       20 

4-        i 

+     84 

a     Aquarii. 

-    I'5 

+     16 

-     15 

74 

-      22 

4-     61 

a     Pegasi  .... 

+      7 

4-      29 

-         2 

-   i* 

'    92 

-       9 

a     Andromcdx     . 

-     73 

+     45 

+       14 

-187 

-    159 

-     76 

12 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 


Corrections  /<>  rig/it  ascensions,  &c. — Continued. 


Star. 

Maskclync,  1805. 

Auwers'  Cacciotorc. 

Oi 

(3 

f-.l" 

ft                     ft 

ft" 

y     Pcgasi  .... 

~     MS 

-   152 

24 

+     56             +52 

+     29 

a     Arietis  .... 

-     I4S 

-  133 

-       5 

—     23              —       8 

-     3i 

a     Ccti        .... 

—     228 
-     Ill 

-  243 
94 

-   "5 
+     34 

+       27                   +12 
+       14                   +31 

—     ii 

+       8 

a     Tauri     .... 

/}    Orionis. 

-     139 

-   159 

-     31 

+     61              +41 

+     18 

ft    Tauri     .... 

-      62 

—    IOI 

+     27 

+   110              +     71 

+     48 

n     Orionis. 

—  123 

-   129 

—       I 

+112                 +    106 

+     83 

ft    Geminoi'um 
(i     Hyclnc 

—  189 
-  133 

—   i/3 

-   161 
—   106 

-   I3& 

-     33 

+      22 

-         S 

+     f'5              +     93 

+     28              +55 

+     25              +62 

+      /o 
+     32 

+     39 

«     Lconis  .... 

ft    Lconis  .... 

-     98 

-     88 

+      40 

+      21                  +31 

+       8 

ti     Virginis 

-    122 

93 

+     35 

29                      o 

-     23 

a     Bootis   .... 

-     97 

-     82 

+     46 

+22            '+     37 

+     M 

ft-   Librge    .... 

50     '          -   101 

+     27 

2                     53                     76 

«     Corona; 

—  1  20 

96 

+     32 

74                     50 

-     73 

«     Serpentis    . 

—  125 

-   148 

—     20 

-     53             -     7G 

-     99 

a     Scorpii  .... 

—  206 

—   223 

-     95 

+     69             +52 

+     29 

a     Herculis 

-  in 

91 

+     37 

Si              -     61 

§4 

a     Ophiuchi     . 

-  163 

-   I3i 

—       3 

28              +       4     |                 19 

a     Lyra      .... 

-     65 

51 

+     77               -       5              +9 

14 

y     Aquilic  .... 

-     7i 

-   in 

+     17 

+     54             +     14 

-       9 

<i     Aquil;c  .... 

-   162 

-   151                      23 

+     44              +55 

+     32 

,3    Aquilse  .... 
a-    Capricorni.     .. 

-  1  20 

-   148 

-   135 
-   126 

+         2 

,+     17              +2 
—      2O                 +2 

—       21 
21 

a     Aquarii. 

-  208     :          -   164 

-      36 

5             +39 

+       16 

«     Pegasi  .... 

-   MO              -   133 

5              +  '47             +54 

+      31 

a     Andromedx     . 

—  146             —   125 

+       3              +23             +44 

+      21 

DETERMINATION   OF   A   SYSTEM  OF   CORRECTIONS. 


Corrections  to  right  ascensions,  &'f. — Continued. 


Star. 

Bessel  I,  1815. 

Bcsscl  II,'i825. 

ft 

Ct 

ft" 

fa 

<•«' 

<"u" 

y     Pcgasi  .... 

+  140 

+    151 

+     98 

0 

+      22    ' 

+      27 

a     Arietis  .... 

+   123 

+    I2O 

+     67 

0                             17                              12 

n    Ccti 

+   126 

+    IOO 

+     47 

o             -     35 

-     3" 

a    Tauri     ....             +20 

+    27 

26 

o                      o 

+       5 

,j    Orionis.      ...             —       S 

-.   33 

-     86 

0 

-    30 

-     25 

,i    Tauri     .... 

+     32 

+     10 

43 

o 

-        9 

4 

«     Orionis.      .      .      .                    45 

5i 

-   104 

0 

-       3 

+         2 

/j    Geminorum      .      .                       5 

J 

51 

o 

17 

12 

«     llydne  ....                   78 

—     8° 

—   I3C 

o 

—     28 

00 

a     Lconis  ....                     16 

—          2 

j  j 

-     55 

o 

4 

^  J 

+     I 

;i    Lconis  .      .      .      .   1         +34 

+       21 

-     32 

0 

31 

—    26 

'(     Virginis 

-       3 

+        M 

39 

0 

+   17 

+      22 

a     Bootis  .... 

+     33 

+     4° 

13 

0 

-       8 

-       3 

+   166 

4-    I3O 

+     86 

o 

_       s 

T 

a     Coronx       ... 

+     102 

i       *  Jy 

+    ICQ 

+     56 

0 

6 

J 

a     Serpentis    . 

+     83 

+     73 

+     20 

0 

+       I 

+       6 

a     Scorpii  .... 

+     121 

+   118 

+     65 

0 

+       7 

+       12 

a     Hcrculis 

+     53 

+     66 

+     13 

o 

+       7 

+       12 

a    Ophiuchi    . 

+     27 

+       20 

-     33 

0 

-     37 

-      32 

a     Lyric      .... 

+     56 

+     61 

+       8 

0 

—         2 

+         3 

j     Aquilx  .... 

+     65 

+     44 

9 

o 

-       6 

I 

a     Aquilx  .... 

+     48 

+     55 

+         2 

0 

+       4 

+         <) 

t3    Aquilie  .... 

+     63 

+     7i 

+     18 

0 

+     26 

+      31 

n-   Capricorni. 

+     52 

+     76 

+    23 

0 

+     26 

+      31 

a    Aquarii. 

+     (>4 

+     37 

16 

o 

+      13 

+     18 

«     Pcgasi  .... 

+    112 

+  116 

+    63 

0 

+         2 

+       7 

«    Andromeda    . 

+    123 

+  123 

+    70 

o 

17 

—       12 

RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 


Corrections  to  right  ascensions,  &c. — Continued. 


Qt.-l  Y 

Struve. 

Brinkley. 

Mar. 

cr 

c-,' 

CT' 

<8     ' 

<"»' 

<•»" 

7     Pegasi  .    •  .      .      . 

+     II 

+     35 

I 

-     150 

—   126 

+       9 

a     Arictis  .... 

+     57 

+     33 

+         2 

-     93 

—    112 

+     23 

a     Ccti 

-f-     73 

+     37 

+         I 

-   150 

—  1  86 

51 

a     T.iuri     .... 

+      22 

+      21 

-       15 

-   145 

-   146 

—      it 

ft    Orionis. 

+  119 

+      89 

+     53 

-     82 

-    112 

+     23 

,?     Tauri     .... 

+     20 

+      14 

—       22 

-   152 

—  160 

—     25 

a     Orionis. 

+   28 

+      22 

-       14 

-   128 

•-  134 

"+        I 

/?    Geminorum 

+     34 

+      13 

-       23 

-     So 

-     101 

+     34 

+    03 

+     61 

+       25 

-       6 

-     3? 

+      Q7 

a     Lconis  .... 

1      j  j 

+     39 

+      32 

-   137 

-   M4 

1        j  i 
-       9 

,:)     Leonis  .... 

+     58 

+     24 

—       12 

-   no 

-  144 

9 

a     Virginis 

+       4 

+     20 

16 

-  182 

-   1  66 

3i 

a     Bootis   .... 

+     29 

+    19 

-     17 

-  139 

-   149 

M 

4-'     22 

+     17 

—       10 

—  126 

—   131 

+       4 

a     Cororue 

+     73 

+     66 

j 

+     30 

-  137 

j  * 
-  144 

—       9 

a     Scrpcntis    . 

+     71 

+     74 

+     33 

-  156 

-  153 

IS 

a     Scorpii  .... 

+     44 

+     52 

+     16 

—  1  86 

-  177 

42 

a     llcrculis 

+     67 

+     74 

+     38 

-   115 

—   108 

+     27 

«     Ophiuchi    . 

+   118 

+     76 

+     40 

-     Si 

—   123 

+       12 

tt       L)TDB        .... 

-       6 

-       9 

-     45 

-    101 

-  104 

+     31 

}'     AquiUe  .... 

+     50 

+     46 

+       10 

-  146 

-  150 

-     15 

«     Aquilic  .... 

+     27 

+     30 

—       6 

,-  MS 

-  MS 

10 

,?    AquiUc  .... 

+       10 

+     39 

+       3 

—  205 

-   176 

41 

<i~    Capricorn!. 

+     37 

+     63 

+     27 

—   189 

—  163 

-     28 

a     Aquarii 

+      21 

+     33 

-       3 

-  127 

-   "5 

+     20 

a.     Pegasi  .... 

+       5 

+      '7 

-     29 

—  132 

-   130 

+       5 

a    Andromedx    . 

-       0 

-     25 

-     61 

-     87 

-  106 

+     29 

DETERMINATION   OF   A   SYSTEM   OF   CORRECTION'S. 


Corrections  to  right  ascension,  &c. — Continued. 


Star 

Argelander. 

I'ulkowa. 

fa 

f» 

Ct" 

Cut 

c' 

fu>" 

Y    Pcgasi  .... 

—        1C 

-4-      11 

—         CQ 

—     ii 

—       6 

«     Arietis  .... 

*  ;? 
+      2S 

1^         **T 

+         2 

DV 

+       62 

+       8 

+     13 

n     Cell  

+      26 

—       ic 

4-       c  i 

o 

-U           c 

*  j 

1                J~T 

^           D 

«     Tauri     .... 

+       12 

+       8 

t                f 

+       6 

16 

—        II 

ft    Orionis. 

+       9 

-     23 

. 

+     3S 

+         2 

+       7 

,1    Tauri     .... 

+     13 

+       12 

• 

-      '7 

+       10 

+     15 

«     Orionis. 

+     17 

+    II 

.    • 

—      13 

-       17 

-       12 

fi    Gcniinorum     . 

+     44 

+   14 

t                t 

+     7° 

+         I 

+       6 

n     Ilydrx  .... 

+     44 

—        I 

• 

+     90 

O 

+       5 

a     Lconis  .... 

+      12 

o 

+     50 

3 

+         2 

/J     Leonis  .... 

+     42 

I 

. 

+     75 

0 

+      5 

n     Virginis 

-     26 

13 

+       3 

+         2 

+       7 

a     Bootis   .... 

+     s 

-       6 

. 

+     38 

+       5 

+       10 

n2    Libra;    .... 

—       8 

4 

—     60 

if) 

—     ii 

«     Corona; 

+     36 

+      22 

+     34 

7 

—         o 

«     Scrpcntis    . 

—          2 

+         6 

. 

-     37 

-       8 

-       3 

«     Scorpii  .... 

-     55 

41 

54 

"9 

M 

a     Hcrculis     . 

+        6 

+       10 

• 

-     ID 

-     23 

18 

n     Ophiuchi     . 

+     73 

+       If> 

. 

+     94 

21 

1  6 

a     Lyrx      .... 

+       6 

O 

+     34 

+       12 

+     17 

}•     Aquihe  . 

-1-       7 

+       10 

31 

+       5 

+       10 

n     Aquilffi  .... 

_       5 

-       3 

-     +       7 

+     I 

+       f) 

/}    Aquihc. 

34 

+       4 

. 

75 

-       5 

0 

a-    Capricorn!  . 

-     5S 

-     31 

—     32 

7 

—         2 

«     Aquarii 

-     26 

21 

+     16 

10 

—       5 

a     Pegasi  .... 

+       7 

+       8 

—     ii 

-     15 

10 

a     Andromeda;    . 

+     41 

+      13 

-t-     63 

+       I 

+       6 

i6 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 


Corrections  to  right  ascensions,  crV. — Continued. 


Star. 

Greenwich,  1864. 

Washington. 

''u 

CM' 

fti" 

Cn  ' 

tii 

Ca" 

7     Pcgasi  .... 

-  124 

51 

-     30 

-     59 

+      22  . 

—       12 

a.     Arictis  .... 

+     53 

—      33 

12 

+  123 

+     27 

-       7 

a     Ceti  

+     53 

13 

+       8 

+  132 

+     5& 

+      22 

a     Tami     .... 

+     40 

+         1                  +22 

+     9i 

•f     48 

+      14 

/?    Orionis. 

+     24 

18 

+       3 

-     73 

+     29 

—       5 

/?    Tauri     .... 

-     58 

o 

+      21 

—     52 

+     16 

-     18 

a     Orionis. 

+       6 

+       3 

+    24 

+     38 

+     35 

+       i 

/3    Geminorum     . 

+  in 

-       3 

+     18 

+  151 

+     23 

n 

a     Hydrne  .... 

+  129 

—     15 

+       6 

+  185 

+     23 

—     n 

a.     Leonis  .... 

+     83 

13 

+       8 

+   148 

+     39 

+       5 

/?    Leonis  .... 

+  no 

4 

+     17 

+  160 

+     33 

i 

«     Virginis 

—      22 

-     38 

17     < 

+     57 

+     37 

+       3 

a.     Bootis   .... 

+     39 

-     17 

+       4 

+     97 

+     33 

i 

03 

—       5 

+     16 

—     67 

+     34 

0 

a     Coronx 

+     48 

23 

—         2 

+     98 

+     17 

17 

a     Serpcntis    . 

-     79 

-     25 

4 

-     32 

+     29 

-       5 

a     Scorpii  .... 

-     91 

-     32 

n 

-     19 

+     48 

+     M 

a     Ilerculis 

+      12 

10 

+     II 

+     62 

+     35 

+       i 

a     Ophiuchi     . 

+  178 

-       5 

+     16 

+  246 

+     42 

+     '8 

n     Lyric      .... 

+       9 

29 

g 

+     68 

+     25 

—       9 

y     Aquilx  .... 

-   117 

46 

-     25 

42 

+     40 

+       6 

«     Aquilx  .... 

-     32 

—     18 

+       3 

+     49 

+     34 

0 

/?    Aquihc  .... 

-   148 

39 

18 

-     89 

+     33 

—       i 

n2   Capricorn  i. 

54 

-     29 

-       8 

o 

+     25 

9 

«     Aquarii. 

+       6 

-     53 

-     32 

+   "5 

+     4<J 

+       12 

«     Pegasi  .... 

—      2O      :                     30 

9 

+     60 

+     49 

+       15 

a     Andromedx     . 

+78                     -22 

I 

+   151 

+     40 

+       6 

DISCUSSION   OF    PROBABLE   ERROR. 


The  following  are  the  values  of  x  and  y,  in  which  the  units  are  thousandths  of 
seconds  of  time: 


Star. 

X 

y 

}•     Peijasi   

-     18 

—  127 

ft     Arietis 

+       12 

F-  167 

«     Ccti  . 

-J-      -12 

-t-    89 

(i     Tauri     

-|-      n-3 

ft    Orionis        .... 
ft    Tauri     .      .      . 

+       2S 

+      M 

+      32 
—    lf'5 

«     Orionis        .... 
ft    Geminorum 
«     llvdnc  
«     Lconis  . 

+       5 
+       10 

+     19 

-       5 
+   236 
+   286 
+   225 

ft    Lconis  .... 

+     24 

+   206 

a     Virginis      .... 
«     B,,otis   . 

if) 
+        4 

+     78 
+    121 

«-    Librae 

-4-        IT 

—    211 

n     Coronic       .... 
n    Serpentis    . 

+           I 

+       3 

+    I  f)0 
-    129 

fi     Scorpii  . 

—       3 

-     I2S 

«     Keren!  is      .... 
"     Ophinclii     .... 
ti     Lvrx 

-       9 
+      2f) 

+       72 

+  357 
+     83 

}'     Aqnil:c 

+      12 

-    1  88 

ft     Aquilrc  ... 

—       5 

+      39 

ft     Aquihc  . 

'            ' 

—       10 

—   206 

«:   C'apricorni 
a     Aquarii        .... 
n     I't^asi 

-     25 

18 

—            3 

I 
+    174 
+     29 

«     Andromcdic     . 

+         II 

+   I'W 

$3- 
DISCUSSION  OF  PROBABLE  KRROK. 

From  these  discordances  we  shall  next  deduce  the  probable  error  of  the  resulting1 
x  and  y.  It  is  well  known  that  results  for  probable  error  generally  prove  entirely 
illusory.  This  does  not  arise  from  any  defect  in  the  theory  of  the  subject,  but  from 
the  fact  that  the  probable  error  generally  has  to  be  deduced  from  the  discordances 
among  a  series  of  results  affected  by  some  common  source  of  error,  which  changes 
;ill  the  results  in  the  same  way,  and  is,  therefore,  not  eliminated  from  the  mean.  15ut 
the  twelve  star-catalogues  here  employed  may  be  regarded  as  absolutely  independent 
of  each  other. 

In  deducing  the  probable  error  from  each  series  of  residuals,  I  take  the  mean 

value  of  the  residuals  without  regard  to  the  sign.     In  the  more  elegant  method   of 

deducing  the  probable  error  from  the  sum  of  the  squares  of  the  residuals,  it  seems  to 

me  that  too  much  weight  is  assigned  to  those  large  residuals  the  presence  or  absence 

3— v  s 


jg  RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

of  which  is  a  mutter  of  chance.  The  probable  error  being  proportional  to  the  mean 
error,  and  only  a  little  smaller,  I  shall  use  the  latter  throughout  instead  of  the  former. 
The  mean  values  of  the  numbers  found  in  the  third  columns,  taken  without  regard  to 
the  sign,  are  as  follows: 

1.  Auwers'  Bradley     ...  10.7 

2.  Piazzi      -      -  49 

3.  Maskelyne    ....  3° 

4.  Auwers'  Cacciotore  36 

5.  Bessol  I  47 

6.  Bessel  II  ^3-7 

7.  Struvo     -      -      -      -  211 

8.  Brinkley  23.3 

9.  Argelander  11.4 

10.  Pulkowa  8.2 

1 1.  Greenwich   .  13.1 

12.  Washington  7-9 

Had  the  catalogues  all  been  entirely  accordant,  these  residuals  would  all  have 
been  zero.  Let  us  next  see  in  what  manner  they  are  made  up.  In  the  case  of  Bradley 
the  residual  is  formed  by  subtracting  from  each  ct  the  value  of  x  —  0.65  y  deduced  from 
the  expression  on  page  10,  substituting  for  ci}  r2,  etc.,  their  values  found  in  the  preceding 
table.  Substituting  in  this  expression  these  values  of  x  and  y  in  terms  of  (\,  cy,  etc., 
'  which  have  already  been  given,  we  find— 

x  —  0.65  y  —  .68  ct  +  .10  c.,  -f  .09  c,,  -f  .09  c4  -f  .06  c~,  +  .14  CR  +  .13  r7 
+  .04  r8  +  .07  rfl  —  .07  r,0  —  .1 2  cn  —  .22  f,2 

Subtracting  this  from  r,,'the  difference  is— 

.32  Ci  —  .  i o  r.,  —  .09  r:,  —  .09  r4  —  .06  Cr,  —  .  1 4  c6  —  .  1 3  c-  —  .04  rH 
-  .07  ca  +  .07  r,0  +  .12  Cu  +  .22  r12  —  r;' 

and  it  is  this  expression  of  which  the  value  is  given  in  the  second  column  under  each 
catalogue  in  the  preceding  tables. 
Let  us  now  represent  by— 

£„  fo,  e,       .  f12 

tlie  probable  errors  of — 

c\i  CM  fa  ....        f12 

or  of  the  star-positions  in  each  catalogue.  The  square  of  the  mean  error  of  a  sum 
being  equal  to  the  sum  of  the  squares  of  the  individual  mean  errors,  the  square  of  the 
mean  error  of  the  last  expression,  or  of  c',  will  be — 

.IO2   £i~  +  .OTO  f.,2  +  .OO8   £:i2  -f  -OOS   £,,2  -f  .OO4   £*  +  .029   £,f  -f-  .OI  7   £-,'  -(-  .OO2    E^ 
+  -005    £,/  +  .005    £102  +  .014   £U2+  .048   £K" 

The  mean  Bradley  residual  being  10.7,  the  value  of  this  expression  is  114. 
A  similar  equation  being  formed  for  each  of  the  twelve  catalogues,  we  have  twelve 
equations  of  the  first  degree,  which  suffice  completely  to  determine  the  twelve  values  of 


DISCUSSION    OF   PROBABLE    ERROR. 


F,  and  therefore  the  mean  error  of  each  catalogue.     These  equations  are  as  follows, 
writing  only  the  co-efficients  of  the  unknown  quantities: 


* 

E 

* 

f6" 

* 

tf 

',',« 

^ 

.IO2 

.010 

.OOS 

.008 

.004 

.O2O 

.017 

.002 

.005 

1 

.005 

.014 

.048 

--  114 

.109 

.866 

.004 

.004 

.OO2   .014 

.014 

.OOI  :   .OO6 

.012 

.O 

.0 

—  2601 

.084 

.002 

.  902 

.004 

.OO2 

.014 

.012 

.001    .OOO 

.017 

.000 

.000 

=  900 

.084 
.036 

.002 
.002 

.OO4 
.002 

.902 
.002. 

.OO2 
.922 

.014 
.OI2 

.012 
.OI2 

.001 
.001 

.0 

.006 

.017 
.032 

.0 

.002 

.0 
.006 

—  1296 
=  2209 

.022 

.002 

.002 

.002 

.OO2 

.792 

.OI2 

.001 

.006 

.040 

.OO5 

.OIO 

=  196 

.020 

.OO2 

.OO2 

.002 

.OO2   .012 

.792 

.001   .006 

.044 

.OO5 

.012 

-  441 

.020 

.002 

.002 

.002 

.002   .OI2 

.OI2 

.940 

.006 

.044 

.005 

.012 

=  529 

,OI2 

.001 

.001 

.OOI 

.002   .012 

.OI2 

.OOI 

.846 

.048 

.OO6 

.017 

=  121 

.OOI 

.000 

.000 

.OOO 

.OOI 

.OIO 

.OIO 

.001 

.006 

.504 

.017 

.044 

=   67 

.029 

.000 

.000 

.OOO 

.OOI 

.008 

.OIO 

.001 

.006 

-'37 

.672 

.090 

-  170 

.048 

.'  0 

.000 

.OOO 

.OOO 

.ooS 

.OIO 

.OOI 

.006 

.152 

.036 

.462 

=   63 

Those  equations  will  give  the  middle  scries  of  probable  errors  with  great,  certainty  ; 
but  those  which  lie  near  the  extremes  in  point  of  time  will  be  quite  uncertain,  the 
necessary  uncertainty  of  the  last  members  of  each  equation  being  considerably  exag 
gerated  iu  the  result.  If  we  solve  by  successive  approximation,  the  equations  for 
determining  e*  and  F,./  will  be — 

.  1 02  e i~  zr  1 1 4  —  74  —  40  ;   f ,  =.  o".O2O 

.462  F12a  =    63  —  55  =    8  ;   F,,  —  o*.oo4 

We  may  roughly  estimate  the  uncertainty  of  the  second  members  of  the.  preceding 
equations  as  equal  to  one-fifth  of  their  entire  amount,  so  that  the  numlx?r  for  which  we 
have  found  the  value  1 14  might  have  ranged  anywhere  between  91  and  135,  while 
that  which  we  have  found  to  be  63  might  have  ranged  between  50  and  76.  Hence  the 
value  of  F!  may,  from  this  cause  alone,  range  between  os.oi3  and  o".O25,  and  that  of  F,., 
between  zero  and  O8.oo7.  In  the  latter  case  we  have  an  independent  means  of  esti 
mating  the  probable  error.  The  catalogue  result  actually  employed  is  the  mean  of 
two  nearly  independent  determinations  with  different  instruments.  The  mean  differ 
ence  .of  the  results  was  o".oi3.  Wo  may  therefore  put  o".oo6  as  the  inferior  limit  of  £,„. 
To  make  the  estimate  too  great  rather  than  too  small,  I  have  supposed  e12  to  be  o*.oo8, 
and  hence  obtained  the  following  scries  of  probable  errors,  which  nearly  satisfy  all  the 
equations  except  the  last: 


X. 

F!  —.019 
^  =-054 

f.  =-037 
F,,  =  .048 
F(i  —.014 
F7  =.023 

<H  =-023 
F,,  =.010 
F10  =  .OIO 


tf  =     36l 

!•./  ZZ29I6 

f:1-  =1      9OO 

ff  =1369 

f,-  —  2304 

f"ti~  =     '96 

v  =529 

529 
100 
100 


f/  =: 


f—  — 
ID  — 


F,.,  —  .OO8 


?n"=    196 

',.;-  =    64 


20  RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

It  may  be  remarked  that  the  only  uncertainty  among  these  probable  errors  which 
materially  affect  our  final  result  for  the  probable  error  of  the  concluded  positions  is  that 

of    £(. 

To  find  the  probable  error  of  a  computed  star-position  at  the  time  t,  we  must  sub 
stitute  the  values  of  x  and  y  in  the  expression  x-\-  y  t,  which  gives  the  correction  to  the 
Tubitlce  licyiomontance  at  the  time  t.  Calling  this  correction  ^  we  have  — 

-4  =  +  (.17  -'.789*1 

+  (.04  —  .I00<3 

+  (.04  -.08  f)c3 


+  (.04  —  .03  0  c5 
+  (.n  —  .  040  A; 
+  (.  1  1  —  .03  0  c7 
+  (.03  —  .01  Of« 
+  (.o8  +  .oiQcb 

+  '(.19  +  .409  CM 
4-  (.06  +  .  279  eu 

+  (-09  +  479cis 

The  square  of  the  probable  error  of  this  expression  will  be  — 

(.  1  7  -  .78  O2  *,s  +  (-04  -  -  iotf  v  +  etc.  =  36  -  1  25  1  +  303  f 

t  being,  it  will  be  remembered,  the  fraction  of  a  century  after  1820.     This  expression 
would  attain  its  minimum  in  1840,  when  its  value  would  be  — 

£~  =  23  or  s  —  os.oo48 
At  any  other  time  T  its  value  will  be  — 


/T-i84oV 
£  ---  23  4-  1,01,  I 

r  °  °  y      100     y 


Owing  to  the  independence  of  the  twelve  determinations  on  which  the  concluded 
positions  rest,  I  conceive  that  this  expression  for  the  probable  error  is  not  entirely  illu 
sory,  but  is  very  nearly  real.  Of  the  exactness  of  the  constant  term  no  serious  doubt 
can  be  entertained.  But  the  co-efficient  303  multiplied  by  the  square  of  the  time 
depends  very  largely  on  the  concluded  probable  error  of  Auwers'  Bradley,  and  we 
have  seen  that  there  is  some  uncertainty  in  this  determination.  If  this  probable  error 
is  really  as  great  as  os.O25,  which  we  have  seen  to  be  quite  possible,  we  should  have— 

/T-i84iy 
t-—  23  +  464  (  —  1 

h  y      100      / 

The  first  expression  will  give  for  the  probable  error  in  1900— 

£  zz  os.o  r  i 
while  the  second  will  give  — 

£  ZZOa.OI4 

It  must  be  understood  that  these  results  for  probable  error  are  true  only  on  the 
assumption  that  the  proper  motion  of  each  star  carries  it  forward'  on  a  great  circle  with 
a  uniform  velocity. 


SYSTEMATIC    DISCORDANCES   AMONG   THE   CATALOGUES. 


21 


$4- 
SYSTEMATIC  DISCORDANCES  AMONG  THE  CATALOGUES. 

Let  us  next  inquire  how  far  the  outstanding  residuals  of  the  corrections  to  each 
catalogue  are  of  a  systematic  character  depending  either  upon  the  right  ascension  or 
declination.  1  have  already  set  forth  the  reasons  for  believing  that  the  systematic, 
errors  to  be  looked  for  are  those  depending  on  the  sine  and  cosine  of  the  right  ascen 
sion.  Let  us  then  suppose  the  residuals  of  each  catalogue  to  be  of  the  form— 

m  cos  a  -f  n  sin  a  -f-  c 

If  we  put  >•,,  r.,,  etc.,  for  the  residuals  corresponding  to  the  several  stars  Ave  shall  have 
twenty-seven  equations  of  the  form— 

m  cos  oci  -f-  n  sin  <*;  +  c  —  r-t 

from  which  to  determine'  -m,  11,  and  c.  If  these  equations  be  solved  by  least  squares, 
the  normal  equations  will  be — 

m  2  cos2  a-\-n  2  sin  a  cos  a  -f-  c  2  cos  a  —  2  r  cos  a 
m  2  sin  a  cos  a  -(-  n  2  sin2  a  -\-  c  2  sin  a  —  2  >•  sin  a 
m  2  cos  ct-\-  n  2  sin  a.  -f-  2  7  c  —  2  >' 

( )wing  to  the  minuteness  of  the  quantities  2  cos  <x,  2  sin  <*,  w,  and  11,  and  to  the  fact 
that  when  we  take  the  residuals  r;!  in  the  third  column  of  the  table  of  catalogues  we 
have  2  r  —  o,  we  may,  using  these  same  residuals,  determine  m  and  n  from  the  first 
two  equations,  putting  c  =  o.  Putting  for  2  cos2  a-,  etc.,  their  values  derived  from  the 
right  ascensions  of  the  twenty-seven  stars  in  question  at  or  near  the  mean  epoch,  these 
equations  become — 

1 1 . 74  m  +   0.93  n  —  2  >•  cos  a 
0.93  m  +  15.26  n  -=.2  r$\\i  a. 
which  give — 

i  i  .68  m  —  2 r  (cos  a  —  .06  sin  a) 
i  5. 1 9  n  —  ^  r  (sin  a  —  .08  cos  a) 

Computing  the  values  of  m  and  n  corresponding  to  each  catalogue  from  these  formula.', 
we  find  that  the  residuals  in  the  second  column,  which  represent  the  excess  of  the  cat 
alogue  right  ascension  over  that  computed,  nuiy  be  represented  as  follows: 


No. 

Catalogue. 

Systematic  residuals. 

s.           s.                     s. 

i 

Auwers'  Hradley    . 

r  =  +  .031  —  .003  cos  i  +  .007  sin  a 

0 

IMazxi,  (1800)     .... 

r=  —.087  +.003  cos  a  —.001  sin  a 

3 

Maskclync,  (1805)  .      .      . 

r=  —.128  —.019  cos  a  —.006  sin  a 

4 

Auwers'  Cacciotore 

r  =  +  .023  +  .010  cos  a  +  .033  sin  a 

5 

Hessel  I,  (1815).      .      .      . 

r  —  +  .053  4-  .037  cos  a  —  .037  sin  a 

6 

Hessel  II,  (1825)      .      .      . 

)•  =;  —  .005  +.006  cos  a  —.010  sin  a 

7 

Slruvc,  (1825)     .... 

r  =  +  .036  —  .007  cos  a  —  .005  sin  a 

S 

Hiinklcy,  (1825  .      .      .      . 

/•=  —.135  —.002  cosrt  +.oot)  sin  a 

g 

Argelandcr,  (1828).      .      . 

r=      .000  +  .001  cos  a  +.002  sin  a 

10 

1'ulkowa,  (1845)     .      .      . 

r  =  —  .005  +  .001  cos  a  +  .002  sin  a 

II 

Greenwich,  (1864)  . 

>-=  —.021  —.oil  cos  a  +.010  sin  n 

12 

Washington,  (1870) 

r  —  +.034  +  .004  cos«  —  .001  sin  « 

RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 


Tlie  catalogues  in  which  the  periodic  character  of  the  residual  is  most  strongly 
indicated  are  Auwers'  Cacciotore,  Bessel  I,  and  Greenwich.  In  the  case  of  the  latter 
the  systematic  difference,  small  as  it  is,  makes  up  the  greater  part  of  the  residual,  since, 
if  it  is  subtracted,  the  mean  value  of  the  residual  is  reduced  from  .0131  to  .0064,  while 
the  probable  error  is  reduced  in  an  even  larger  proportion.  Taking  away  the  three 
catalogues  just  mentioned,  the  co-efficients  m  and  n  are  no  greater  than  might  result 
from  the  accumulation  of  accidental  errors ;  it  is  therefore  doubtful  whether  we  ought 
to  regard  them  as  real. 

To  find  what  discrepancies  depending  on  the  declinations  arc  indicated,  I  have 
grouped  the  twenty-seven  stars  as  follows: 

7  stars  north  of  +  20° 
1 1  stars  between  -f-    6°  and  -f-  16° 
9  stars  south  of-f    6° 
The  mean  residual  from  the  third  column  corresponding  to  each  group  is — 


Catalogue. 

North  group. 

s. 

Bradley       .... 

+    .006 

—    .  046 

Maskelyne 

+     .021 

Cacciotore. 

+     .OO5 

Bessel  I      .... 

+     .013 

Bessel  II    .... 

—     .  OOf) 

—       OIQ 

Brinkley     .... 

+     .OIO 

Argelantlcr 

+   .008 

+   .008 

Greenwich.      . 

+   .003 

Washington 

—   .008 

Middle. 

South. 

s. 

s. 

.000 

—    .006 

-    .013 

+  .049 

.000 

—    .022 

+    .002 

—    .006 

—    .  006 

-   .004 

+    .001 

+   .  004 

+    .004 

+     .OIO 

—    .  002 

—   -005 

+    .007 

-    .016 

—    .005 

—   .001 

+    .004 

—   .  006 

+    .003 

+  .003 

Notwithstanding  their  general  minuteness,  these  residuals  depending  on  the  decli 
nation  are  more  certainly  indicated  than  those  depending  on  the  right  ascension.  The 
catalogues  in  which  they  are  largest,  those  of  Piazzi  and  Maskelyne,  have  received 
but  a  small  weight  in  the  preceding  disciission.  In  all  the  others,  with  the  possible 
exception  of  Struve,  they  are  so  small  that  they  may  be  treated  as  accidental  errors. 
Their  farther  consideration  is  therefore,  for  the  present,  omitted. 

§5-  . 

DETERMINATION  OP  THE  POSITION  AND  MOTION  OF  THE  EQUINOX. 

For  the  determination  of  the  common  correction  to  the  right  ascensions  of  all  stars 
on  account  of  error  in  the  assumed  equinoctial  point,  we  are  still  dependent  entirely  on 
observations  of  the  sun.  We  may  ultimately  hope  to  determine  some  of  the  quantities 
on  which  the  absolute  positions  of  the  stars  depend  from  observations  and  theories  of 
the  planets,  the  orbit  of  each  planet,  when  corrected  for  perturbations,  being  a  -fixed 
great  circle,  and  the  planets  being  susceptible  of  more  accurate  observation  than  the 
sun,  but  this  course  is  not  possible  at  present.  It  is  well  known  that  the  best  deter 
minations  of  the  .equinox  exhibit  discordances  far  larger  than  can  result  from  purely 


DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE  EQUINOX. 


accidental  errors  of  observation,  and  HO  indicate  the  existence  of  systematic  errors  in 
the  observations  of  the  sun,  the  nature  and  cause  of  which  are  obscure.  The  deter 
mination  of  each  co-ordinate  of  the  sun  seems  about  equally  liable  to  them.  We  know 
that  the  absolute  right  ascension  of  a  star  depends  effectively  upon  two  elements,  to  be 
determined  from  observation,  namely: 

1 .  The  moment  at  which  the  sun  crossed  the  equinox,  from  which  its  absolute 
rig-lit  ascension  at  any  time  may  be  obtained,  the  elements  of  its  orbit  being-  known. 

2.  The  difference  of  right  ascension  between  the  sun  and  a  star,  from  which  the 
absolute  right  ascension  of  the  latter  is  obtained  from  that  of  the  former. 

Since  any  error  depending  on  the  sun's  zenith  distance  will,  in  the  first  determina 
tion,  be  eliminated  from  the  mean  of  the  t\vo  equinoxes,  it  might  be  supposed  that 
there  would  be  a  general  agreement  among  different  authorities  in  this  determination. 
Hut  such  is  by  no  means  the  case,  and  I  can  assign  no  probable  cause  for  the  observed 
discrepancies. 

With  reference  to  the  second  determination,  it  is  well  known  that  the  personal 
equation  of  an  observer  is  frequently  different  for  the  sun  and  for  a  star,  and  this  dif 
ference  will  appear  in  the  difference  of  their  right  ascensions.  Diurnal  changes  in  the 
clock,  the  instrument,  and  the  personal  equation  of  the  observer  will  also  introduce 
errors  into  the  determination. 

Under  these  circumstances  the  most  we  can  do  is  to  base  our  determination  on 
the  widest  range  of  material  at  our  disposal,  depending  for  accuracy  on  the  number 
and  variety  of  the  determinations  employed  rather  than  upon  the  precision  of  any  one. 
All  the  data  at  my  disposal  which  did  not  involve  a  re-reduction  of  observations  are 
comprised  in  the  following  discussion. 

The  different  common  corrections  to  the  preceding  determination  of  right  ascen- 
siovis  of  stars  are  represented  by  c.  We  begin  with  an  examination  of  those  published 
catalogues  in  which  the  star-positions  are  referred  directly  to  the  sun. 

BESSEL'S  BRADLEY. 

The  right  ascensions  of  the  following  stars  were  determined  by  Bessel  by  direct 
comparison  with  the  sun.  Opposite  each  star  is  given  Auwers'  correction  to  Bessel's 
position,  and  the  concluded  correction  already  found,  the  signs  being  changed: 


Star. 

H.  -  A. 

n  -  (i). 

+   .012 

4-     063 

/?    Orionis     .... 

-   .040 

—    .007 

ra     Orionis     .... 

—   .030 

—    .008 

f)    Gcminorum   . 

+    .102 

+    .143 

a     Lcnnis       .... 

+   .  106 

+    -149 

n     Virginis    .... 

+    .042 

+    .070 

-4-      036 

+    .075 

a     Lyrnc    

+  .030 

+    .058 

a     Aquilx      .... 

+   .012 

+    .030 

Mean     .... 

+  .030 

+    .064 

RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

We  liavo,  therefore,  from  Bessel's  determination  for  1755— 

c  =z 


PlAZ/I,    l8oO. 

Of  Piazzi's  determination  of  tlie  equinox  I  know  nothing-.  The  mean  correction 
given  by  his  catalogue  cannot,  therefore,  receive  much  weight,  although  the  geograph 
ical  position  of  his  observatory  is  very  favorable  to  its  determination.  The  value  of  c, 
resulting-  from  this  and  from  all  the  other  original  catalogues  employed  in  the  preceding- 
determination  is  the  same  with  the  constant  terms  in  the  systematic  residuals,  given  on 
page  21,  and  represented  by  r. 

MASKELYNE,  1805. 

Maskclyne  states  that  his  catalogue  was  "settled  with  respect  to  the  equinoctial 
points,  from  the  observed  declinations  of  the  sun  about  the  equinoxes  in  1804,  1805, 
and  1806."  With  the  same  catalogue  he  gives  one  for  1756,  deduced  from  Bradley's 
observations,  the  mean  of  which  is  o'.io  less  than  Bessel's,  if  we  compare  the  twenty- 
seven  stars  under  consideration,  or  o".  1  5  less,  if  we  compare  only  the  stars  which  Bessel 
found  by  direct  comparison  with  the  sun.  This  large  difference  may  arise  from  Maske- 
lyne's  having  used  only  observations  of  the  sun  very  near  the  equinoxes,  while  Bessel 
extended  his  determination  to  a  declination  of  14°  on  each  side.  An  examination  of 
the  tables  of  corrections  corresponding  to  different  declinations  of  the  sun,  given  on 
page  15  of  the  Fundamenta,  docs,  in  fact,  show  a  systematic  difference  depending  on 
the  declination,  but  their  source  and  nature  are  too  obscure  to  admit  of  much  discus 
sion.  All  we  can  do  is  to  suppose  Maskelyne's  determination  an  independent  one, 
entitled  only  to  small  weight. 

BESSEL,  1815  AND  1825. 

Both  Bessel's  catalogues  are  referred  directly  to  the  sun.  The  first  depends  on 
three  years'  observations  with  the  old  transit  and  the  Cary  circle,  numbering  290  in 
all;  the  second  on  five  years'  observations  with  the  Reichenbach  meridian-circle,  num 
bering  about  400  in  all. 

STEUVK,  1825. 

Struvc's  equinox  depends  on  216  observations  of  the  sun,  extending  through  a 
period  of  four  years.  (Positiones  Media1,  p.  xxxx.) 

ARGELANDER,  1830. 

From  a  remark  in  the  introduction  to  his  catalogue,  page  iii,  it  would  appear  that 
Argelander,  from  his  own  observations  of  the  sun,  found  a  correction  of  +©".072  to 
the  right  ascensions  of  the  Tabulae  llegiomontana?.  We  have,  therefore,  on  this 

authority— 

c—  +  o".o64;  epoch,  1828. 

POND,  1830. 

• 

In  the  introduction  to  his  "Catalogue  of  1112  stars,"  Pond  states  that  the  right 
ascensions  of  Maskelyne's  last  catalogue  have  been  increased  by  o'.2O,  being  a  quantity 
determined  by  observations  of  the  sun  made  about  the  equinoxes  with  the  present 


DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE   EQUINOX.  25 

transit-instrument  since  1816.  The  mean  value  of  e  given  by  this  catalogue  is 
-f-os.o/3.  Hut,  as  the  mean  epoch  of  the  observations  is  about  seven  years  earlier,  we 
may  inquire  whether  the  correction  is  different  for  this  epoch.  A  correction  of  -f  os.2O 
applied  to  Maskelyne's  catalogue  for  i8o5  would  leave — 

I  infer,  therefore,  that  the  correction  should  have  the  same  value  for  the  mean  epoch, 
which  we  may  consider  to  be  1825. 

AIKV.S  GREENWICH  CATALOGUES. 

The  five  catalogues  published  by  Airy  have  been  so  carefully  referred  to  the  sun 
that  they  do  not  appear  to  need  any  further  discussion.  The  comparison  of  the  indi 
vidual  catalogues  with  the  positions  already  found  give  the  following  values  of  e: 

s. 
First  six-year  catalogue  .    epoch  1839,  r  —  —  .052 

Second  six-year  catalogue  l&45,  e  —  —  .003 

Third  six-year  catalogue  1 85 1,  e  —  -}-  .01  7 

First  seven-year  catalogue  1 857,  i-  —  —  .005 

Second  seven-year  catalogue  .                              1864,  c  —  —  .02  i 

PULKOWA. 

The  Pulkowa  equinox  depends  on  observations  of  the  sun  with  the  transit-instru 
ment  and  the  vertical  circle,  extending  from  the  year  1842  to  1850.  The  copious  dis 
cussion  of  these  observations  bv  Wagner  is  found  in  "Observations  de  Poulkova," 
vol.  Ill,  pp.  41-67. 

Adding  Washington,  1870,  this  completes  the  list  of  original  catalogues  referred 
independently  to  the  sun.  But  we  have  still  a  collection  of  data  in  the  form  of  obser 
vations  of  the  sun  made  and  reduced  at  various  observatories,  and  compared  with  the 
tables.  I  have  reduced  all  these  in  a  sumniarv  manner,  which,  though  coarse  and  rude, 
will,  I  am  persuaded,  lead  to  results  differing  from  those  of  the  most  refined  discussion 
only  by  quantities  much  smaller  than  the  unavoidable  probable  errors  of  the  results. 
The  mean  correction  to  the  tabular  right  ascension  and  declination  was  taken  for  each 
month  through  a  series  of  years,  all  the  observations  made  in  any  month  of  a  given 
name — January,  for  instance — being  combined  into  a  single  mean.  This  mean  was  that 
of  the  entire  number  of  individual  results.  The  correction  to  the  sun's  absolute  mean 
right  ascension  was  then  deduced  from  the  observed  errors  of  the  tabular  declination, 
being  given  by  the  approximate  formula — 

S  R.  A.  (in  time)  — .'.  2  A  <5  cos  / 

/  being  the  sun's  longitude,  and  the  factor  cos  /  being  the  meail  value  of  the  cosine  of 
the  sun's  true  longitude  during  each  month. 

The  mean  correction  of  the  sun's  absolute  right  ascension  thus  obtained  is  com 
pared  with  the  correction  obtained   by  comparison  with   the   stars,  and  the  difference, 
supposed  to  be  due  to  error  in  the  adopted  right  ascensions  of  the  latter,  gives  the  mean 
correction  to  the  equinox  to  obtain  the  absolute  right  ascension  of  the  stars. 
4— F  s 


26 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 
PARIS,  i854-'63. — (Corrections  to  CarfniUs  tables.} 


No  of 

No.  of 

Month. 

A  a 

obs. 

<5 

obs. 

cos  / 

<5  cos  / 

s. 

,, 

.. 

January     . 

+      .132 

56 

+     1.07 

43 

+        -44 

+     0.47 

February  . 

+      .154 

86 

+     2.76 

73 

+        .Si 

+     2.24 

March  . 

+      .165 

104 

4-     2.  68 

98 

+        .98 

+     2.63 

Apr  1    .      .      . 

+      .209 

116 

+     2.25 

105 

+        .89 

+       2.OO 

May     .      .      . 

+      .245 

So 

+     1.73 

77 

+        .56 

4-     i.  oo 

unc     . 

+      -234 

Si 

+      1.20 

74 

+          .10 

4-       0.12 

July      .      .       . 

+      .207 

1  20 

+     o.So 

84 

•39 

•     0.31 

August 

+      .213 

126 

-t-     0.19 

95 

-       -79 

-   0.15 

September 

+      -194 

98 

0.31 

76 

-       .98 

+    0.30 

October     . 

+      -133 

93 

+     0.50 

6S 

-       -92 

0.46 

November 

+      .196 

72 

4-     o.n 

52 

-       .60 

—     0.07 

December 

+      .148 

49 

+     1-30 

40 

—       .  10 

•   0.13 

Sum 

+    2.230 

•      • 

+     7-f>4 

We  hence  deduce— 

s. 

Mean  correction  to  sun's  absolute  right  ascension  -(-0.212 

Mean  correction. deduced  from  stars   ...  +0.186 

Therefore  star's  right  ascensions  require  the  correction  +  0.026 

A  comparison  of  the  corrections  to  the  Tabula?  Regiomontanre  adopted  in  the  reduc 
tion  of  the  Paris  observations  with  those  deduced  in  this  paper  show  that  the  latter  are 
greater  by  o".O24.  We  may  therefore  put  as  the  result  of  this  series  of  Paris  obser 
vations — 

c  =r  os.ooo 

PARIS,  i864-'65 — (Corrections  to  Le  Verrier's  tables.") 


\ 

No.  of 

No.  of 

Month. 

A  a 

obs. 

,5 

obs. 

A  '!  cos  / 

s. 

„ 

„ 

January     . 

—      .072 

10 

+       0.20 

7 

+     0.09 

February  . 

—      .060 

7 

+     1-35 

6 

+     i.  II 

March  . 

+      .028 

16 

4-      i.oS 

16 

+      i.  06 

April   . 

—      .030 

24 

+     1-39 

22 

+     1.24 

May     .      .      . 

—      .013 

21 

—     0.05 

2O 

0.03 

June    . 

—      .052 

19 

0.29 

18 

—     0.03 

July     .      .      . 

.064 

14 

-     o.SS 

16 

+     0.34 

August 

—       .022 

13 

+       O.II 

M 

0.09 

September 

—       .OO6 

23 

+     0.52 

25 

-     0.51 

October    . 

—        .022 

'9 

+     0.45 

1  6 

0.41 

November 

+        .037 

15 

1.48 

14 

+     0.89 

December 

-        .017 

13 

-     o.S6 

13 

+     0.09 

Sum 

-    0.293 

% 

+     3-75 

i 

DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE  EOflNOX. 


We  hence  deduce — 

s. 

Mean  correction  to  sun's  absolute  right  ascension  .  .   +0.104 

Mean  correction  deduced  from  stars  —  0.024 

Correction  to  adopted  right  ascensions  of  stars  +0.128 

The  Paris  concluded  right  ascensions  for  1864  are  less  than  those  here  deduced 
by  os.oi3.  The  result  of  these  two  years'  work  at  Paris  is  therefore — 

r  =  +  os.  1 1 5 

In  the  following  two  years  nearly  all  the  observations  of  the  sun  at  Paris  were 
made  by  a  single  observer,  whose  results  are  remarkably  different  from  those  of  other 
observers,  as  may  be  seen  by  the  very  large  and  uniformly  positive  corrections  to  the 
right  ascension  and  longitude  of  the  sun,  given  in  the  collected  results  of  the  observa 
tions.  This  error  evidently  arises  from  the  personal  equation  of  the  observer  not  being 
the  same  for  the  sun  and  for  a  star.  Such  differences  are  known  to  exist,  but  they  are 
generally  so  minute  as  hardly  to  admit  of  positive  detection.  We  have,  however,  one 
or  two  cases  of  the  differences  attaining  an  extraordinary  magnitude,  and,  as  they  do 
not  follow  the  usual  law  of  error  in  their  distribution -among  different  observers,  we 
must  reject  entirely  the  cases  in  question. 

CAMBK1DGE,  (ENGLAND.) 

The  results  of  the  observations  of  the  sun  during  the  five  years  1833— '37  are  as 
follows: 


Month. 

da 

No.  61 

obs. 

A,i 

No.of 
obs. 

A  (!  cos  / 

s. 

,, 

,, 

January 

+     .203 

3i 

+      I  .  oo          30 

+     o-44 

February    . 

+     -I'JS 

40     !      +      0.96     ;     41 

+     o.So 

March   .      .      .    •.     -t-      .175 

44           +      ".18      •     44 

+     0.18 

April'    .      .      .    |      +      .230 

55           +     0-36 

56 

+     0.32 

May       .      .      .   i      +      .215 

96 

+     o.f)8 

95 

+     0.38 

June 

+     .138 

71           +      0.35 

77 

+     0.04 

July       .      .      . 

+      .2<>5 

59          4-     0.66 

53 

—     0.26 

August 

+      .182 

77          +     0-14 

80 

•       O.I  I 

Sepicmber  . 

+      .  208 

65     ,     +     0.07 

65 

—     0.07 

October       .      .         +      .180 

73          +     0.46 

75                  <>-43 

November  . 

+     -237 

61           +     0.79           59              •      0.47 

December  .      .        +      .175 

42           +     0.93 

37                  0.09 

Sum 

+  2.343 

• 

•      • 

• 

+     0.73 

Mean  correction  to  sun's  absolute  right  ascension 
Correction  deduced  from  stars     . 
Correction  to  adopted  right  ascension  of  stars  . 
Adopted  right  ascensions  f/rctifcr  than  ours  by    . 

Hence — 

c  —  —  0".03;    epoch,    1835. 


s. 

-f-  O.O2O 
+  0.195 

-0.1/5 
0.146 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 


in  subsequent  years  the  discussion  of  the  correction  for  each  year  in  which  suffi 
cient  observations  of  the  sun  were  made  is  given  along  with  the  results  of  the  obser 
vations.  The  following  are  the  mean  values  of  c  for  series,  each  including  five  years' 
actual  observation: 

,1. 

1841,  r=:-fo.o6 
1849,  c  —  +0.06 
1856,  c:zz  — O.oi 

OXFOED. 

Since  1861  the  sun  has  been  pretty  regularly  observed  at  the  Ifadclifte  Observa 
tory,  and  the  apparent  errors  of  the  tabular  positions  are  given  in  the  results  of  each 
year's  observations.  They  are  divided  into  two  series,  the  first  comprising  the  years 
i86i-'63,  in  which  the  results  are  compared  with  Carlini's  tables,  and  the  observations 
are  considerably  scattered;  the  second,  comprising  the  years  i864~'68,  during  which 
the  comparison  has  been  made  with  Le  Terrier's  tables. 

The  tabular  summary  of  the  results  for  the  years  i86i-'63  is  as  follows: 


No.  of 

No.of 

Month.                     A« 

1 

obs. 

A(i 

obs. 

A  ti  cos  / 

s. 

,, 

,, 

January       .      .        -t-      .18 

9 

+       0.2 

12 

+        O.I 

February     .               +      .40 

'4 

+     0.6 

M 

+   0.5 

March   ...        +      .34 

12 

+    "o  .  6 

15 

+     0.6 

April                 .   !     +     .32 

19 

+      1-4 

19 

+     1.3 

May       .      .      .    ;     +      .33 

16 

+     1.5 

II 

+     0.8 

June       ...         +      .16 

13 

+     0.6 

II 

+       O.I 

July        .      .      .1      +      .30 

1  6 

+     0.5 

17 

—      0.2 

August        .      .         +      .38 

8 

+      1.4 

8 

—      I.I 

September.      .         +      -35 

M 

1-3 

M 

+    1.3 

October       .      .         +      .53 

7 

+     0.8 

9 

-     °-7 

November  .      .         +      .46 

15 

+     0.7 

15 

0.4 

December  .      .         +      .40 

12 

+     2.3 

12 

'—      O.2 

Sum      .      .         +4.15 

.    .     • 

+      2.1 

8, 

Mean  correction  to  sun's  absolute  right  ascension  .  -}-  0.06 

The  same  deduced  from  stars +°-35 

Correction  to  adopted  right  ascension  of  stars    .  — 0.29 

These  adopted  right  ascensions  are  the  same  as  those  used  during  the  correspond 
ing  year  in  the  reduction  of  the  Greenwich  observations,  the  mean  of  which  corresponds 
very  nearly  to  our  adopted  results.  We  have,  therefore,  from  these  Itadcliffe  observa 
tions — 

r  — —  o".29;   epoch,  1862. 

The  great  and  improbable  magnitude  of  this  result  renders  the  question  of  its 
introduction  somewhat  embarrassing. 


DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE   EQUINOX. 

For  the  live  subsequent  years,  i864~'6S,  we  have — 


29 


Month. 

Au 

No.  of 

obs.            A<1 

No.  of 
obs.       A  ''  cos  ' 

s. 

11 

,, 

anuary 

+      .IT 

36            +       2.0 

36       +    0.9 

"ebruary     . 

+        .10 

37          +      i.o 

37          +     o.S 

larch    . 

+    .13 

28     .      +      i.o 

27          +      i.o 

Vpril      .      .      . 

+    .17 

43                 O.I 

45                  o.i 

lay 

+    -15 

45             •     o.i 

47                  o.i 

une 

+      .o3 

56          -     0.7 

55                  o.i 

uly        .      . 

+      .06 

55          +     0.3 

57     i        •     o.i 

August 

+      .08 

47          +     0.7 

47                   0.6 

eptembcr  . 

+      .08 

52         +     o.i 

50      •                 O.I 

)ctober 

+        .10 

46                 •       0.2 

47          -i-     0.2 

November  . 

+       -12 

42          +      I.o 

41                  0.6 

)ecember  . 

+        .10 

+       1.4 

28                  o.i 

Sum 

+     1.28 

'. 

+     I.I 

8, 

Correction  to  sun's  absolute  rig-lit  ascension  -f-  0.03 

Correction  deduced  from  stars  .  -f  o.  1 1 

Correction  to  adopted  right  ascension  of  stars  -0.08 

The  adopted  right  ascensions  beingstill  the  same  as  at  Greenwich,  this  result  gives — 

';  —  —  O'.OQ  :  epoch,  1 866. 

The  correction  is  still  negative,  as  in  the  years  i86i-'63,  though  not  so  great. 

A  comparison  of  the  Radcliffe  results  with  those  derived  from  the  Greenwich 
observations  shows  that  the  difference*  is  found  chiefly  in  the  observed  relative  right 
ascensions  of  the  sun  and  stars.  Tims,  during  the  years  i86i-'63,  the  mean  correction 
to  the  sun's  right  ascension  deduced  from  the  Greenwich  observations  is  -f  o".  17,  while 
that  from  the  Radcliffe  observations,  nsing  the  same  adopted  right  ascensions  of  stars, 
is  +  O8.35,  so  that  the  Uadclifi'e  results  are  greater  by  os.  18. 

An  examination  of  the  Radcliffe  observations  shows  that  this  great  discordance 
did  not  exist  in  the  year  1861,  but  commenced  in  1862,  -\\ith  the  introduction  of  the 
new  transit-circle.  It  might,  therefore,  well  give  rise  to  the  suspicion  that  the  pointing 
of  the  circle  is  affected  with  a  diurnal  change.  As  the  clock-error  is  determined  from 
observations  made  at  night,  a  constant  error  would  thus  be  introduced  into  the  results 
of  transits  of  the  sun.  1'ut  this  explanation  seems  to  be  negatived  by  the  fact  that  the 
observations  of  the  moon,  though  made  at  the  same  time  with  the  stars,  deviate  still 
more  widely  from  the  truth.  In  fact,  the  menu  correction  to  Hansen's  right  ascension, 
given  by  the  observations  of  1862,  is  -j-  o".62,  while  all  other  observations  show  that 
the  right  ascension  was  then  very  nearly  correct.  We  may  heilcc  conclude  that  the 
cause  is  to  be  sought  for  in  some  personal  peculiarities  of  the  observers.  Though  this 


;o 


RIGHT  ASCENSIONS  OF   FUNDAMENTAL   STARS. 


explanation  is  made  unsatisfactory  by  the  circumstance  that  the  observations  of  the 
two  principal  observers  lead  to  nearly  the  same  result,  it  is  noticeable  that  in  the  year 
1863  Ave  find  four  observations  of  the  moon  signed  M.,  probably  made  by  the  director 
of  the  observatory,  the  mean  correction  from  which  is  -j-  oa.O35.  This  seems,  therefore, 
to  be  the  only  admissible  explanation. 

EDINBURGH. 

In  the  Edinburgh  observations  for  the  years  i836-'39  the  correction  to  the  equinox 
is  deduced  from  each  year's  observations.     The  mean  result  is— 

r'^-f-°s-°4j  fpoch,  1838. 

Observations  of  the  sun  were  continued  till    1843.     The  published  corrections  to 
the  tables  lead  to  the  following  mean  results: 


Month. 

<irt 

No.  of 
obs. 

A  6 

No.  of 
obs. 

A  <!  COS  / 

j. 

January 

+  '    -17 

50 

+     1-3 

32 

4-     0.6 

February    . 

4-      .12 

37 

+     1-3 

41 

+     i.i 

March    . 

+      .09 

53 

+     0.4 

60 

+     0.4 

April 

+      .08 

57 

+     0.5 

50 

+     0.4 

May.      .      .      . 

+      .13 

42 

O.I 

46 

0.  I 

June 

+      .11 

51 

+     0.8 

47 

+       O.I 

July.      . 

+      .09 

37 

—      0.2 

43 

4-     o  I 

August. 

+      .05 

43 

—      0.2 

44 

4-     0.2 

September  . 

+      .08 

44 

I.O 

43 

4-      i.o 

October 

+      .18 

32 

-      0.3 

46 

4-     0.3 

November  . 

+      .05 

32 

+      0.3 

20 

—      0.2 

December  . 

+      .13 

3S 

-      0.3 

41 

0.0 

Sum 

+   1.28 

+     3-9 

s. 

Mean  correction  to  sun's  absolute  right  ascension  -j-°-IO8 

Mean  correction  deduced  from  stars     .      .      .      .      .      .       +  o.  106 

Correction  to  adopted  right  ascensions  -f-  0.002 

The  adopted  right  ascensions  are  os.oi  less  than  those  in  this  paper;  therefore — 

-     r  —  —  os.oi;   epoch,  1842. 

BERLIN. 

•  /. 

The  introduction  to  the  first  volume  of  the  Berlin  observations,  published  by 
Encke,  gives  the  results  of  observations  of  the  sun  from  1839  to  1842.  The  following 
are  the  mean  results: 


DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE  EOUINOX. 


Montli. 

A<; 

N'o.  of 
obs.    .          A'! 

No.  of 
obs.        A  *  cos  ' 

s. 

January 

•  14 

4                      1.4 

5            —     0.6 

February     . 

+      .04 

20                  o.o 

20                   o.o 

March    .      .      . 

+        .12 

5                +0.2                4                +0.2 

April 

4-      .06           IS             +     0.9 

18            +     o.S 

May       .      .      . 

+        .02              14                           0.4 

14                  •      0.2 

June 

.00            7                    i.i 

7                    o., 

July        .      .      . 

—    .03 

7            +     2.4 

6               -0.9 

August. 

+       .  12 

II                 +0.1 

II                              0.  I 

September  . 

+    .07 

25                   o.o 

23                  o.o 

October 

+    .04 

13        -    1.3 

10                +       1.2 

November  . 

+    .11 

14        +   0.7 

14                    0.4 

December  . 

+     .11 

7                +       0.2 

S                    o.o 

Sum      .'    . 

-t-    -52 

—       0.  I 

81 

Mean  correction  to  absolute  right  ascension  —  o.oo? 

\J 

Correction  from  stars -f-  0.043 

Correction  to  right  ascension  of  stars  -  0.046 

The  adopted  right  ascensions  are  those  of  the  Tabula;  Regiomontanae,  which  at 
this  time  are  o".oi4  less  than  those  deduced.     We  have,  therefore, 

r  =  —  o".o6;   epoch,   1841. 

The  Berlin  observations  are  distributed  among  the  months  with  unusual  irre«>-u- 
larity.     Had  the  mean  by  weights  been  taken,  the  result  would  have  been — 

(•  —  —  os.o; 

WASHINGTON. 

The  following  are  the  mean  results  of  observations  made  with  the  transit-instru 
ment  and  mural  circle  during  the  four  years  i862-'65: 


No.  of 

No.  of 

Month. 

An 

obs. 

A,! 

obs. 

A(!  cos  / 

January 

—  0.052 

42 

+  0.17 

34 

+  0.07 

February     . 

-  0.045 

42 

—  0.30 

30 

-   0.24 

March    .      .      .        —  0.022 

41 

+  0.40 

31 

+  0.39 

April 

+    O.022 

41 

+  0.70 

36 

+  0.62 

May.      .      .      . 

—    O.O2O 

53 

+    0.22 

53 

+    0.12 

June                   .    [     +  0.008 

50 

+  0.25 

66 

+    0.02 

July        .      .      .    |     +  0.015 

5i 

+  0.30 

55 

-    0.12 

August  ...         -   0.002 

57 

o.oo 

50                o.oo 

September  .      .         —  0.038 

42 

—    0.22 

47 

+    0.22 

October       .      .         -  0.033 

35 

-    0.  10 

55 

+    0.09 

November  .      .         —  0.027 

35 

—    O.  IO 

46 

+    O.o6 

December  .      .         —  0.052 

27 

-r   0.57 

32 

-    0.06 

Sum            .   1      -  0.238 

• 

+    I.I7 

32 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 


Mean  correction  to  sun's  absolute  right  ascension  -f-  0.032 

Correction  deduced  from  stars  -  0.020 

Correction  to  adopted  rig-lit  ascension  of  stars  +  0.052 

The  adopted  right  ascensions  are  those  of  Dr.  Gould,  the  mean  of  which,  at  this 

epoch,  scarcely  differs  from  that  of  our  deduced  positions.  We  have,  therefore,  from 
these  observations — 

f  — +  os.O5;  epoch,  1864. 

Since  1866  the  observations  have  been  made  with  the  transit-circle,  but  the  instru 
ment  was'dismounted  a  part  of  the  time.  The  first  two  years'  work  gave  the  discordant 
results — 

.S'. 

1866  frr-f-0.07 

1867  p—      o.oo 
Mean                          c  —  +  0.035 

The  following  are  the  mean  results  for  i868-'69: 


Xo.of 

No.  of 

Month. 

A  a 

obs. 

A  .5 

obs. 

A  6  cos  / 

s. 

,, 

,, 

January     . 

+      o.  10 

10 

+-       o.S 

8 

+       0.4 

February  . 

+      0.04 

20 

+       0.3 

'9 

+         0.2 

March  . 

+      0.05 

20 

+       0.7 

18 

+         0.7 

April    . 

+      0.06 

15 

+       0.7 

T9 

+       0.6 

May     .      .      . 

4-       O.O2 

22 

o.o 

24 

0.0 

June    . 

4-     o  .  06 

13 

0.2 

13 

o.o 

July     .      .      . 

+    0.07 

15 

-         0.3 

15 

+          O.I 

August 

+     0.06 

9 

+         0.5 

7 

0.4 

September 

+    0.03 

9 

0.7 

9 

+       0.7 

October     . 

—       O.O2 

13 

i.5 

14 

+        1-4 

November 

+       O.OI 

12 

0.7 

12 

+       0.4 

December 

+     o  .  06 

10 

0.9 

IO 

+          O.I 

Sum 

-I-       o    m 

+       4-2 

<^        "•  J*t 

S. 

Mean  correction  to  sun's  absolute  right  ascension  -f-  o.  1 1 7 

Correction  deduced  from  stars  -f-  0.045 

Whence — 

c  —  -f-  os.o7 

The  comparison  is  with  Ilansen's  tables,  the  mean  right  ascension  of  which  is 
about  os.o6  or  os.o7  less  than  that  of  Le  Verrier's  tables. 

The  true  value  of  c  being  of  the  form  .r'-f  ?/T,  every  separate  determination  of  its 
value  gives  an  equation  of  condition  between  x'  and  y'.  The  following  is  the  system 
of  equations  thus  obtained.  I  have  included  the  mean  result  of  the  Greenwich  observa 
tions  for  1868,  1869,  and  1870,  which  have  been  published  since  the  appearance  of  the 
last  Greenwich  seven-year  catalogue.  The  Radcliffe  resiilts  are  omitted. 


DETERMINATION  OF  THE  POSITION  AND  MOTION  OF  THE  EQUINOX. 


33 


The  weights  have  generally  been  assigned  with  reference  to  the  entire  number  of 
observations  on  which  each  result  depends,  and  also  on  the  number  of  years  over 
which  the  observations  extend.  But  they  have  not  been  supposed  to  increase  propor 
tionally  to  either  the  time  or  the  number  of  observations,  so  that,  although  the  weight 
i  would  be  assigned  to  the  result  of  a  single  average  year's  work  with  the  best  instru 
ments,  the  weight  assigned  to  seven  consecutive  years'  work  by  the  same  observers, 
and  with  the  same  instruments,  is  much  less  than  7.  On  the  same  principle,  greater 
proportional  weight  has  been  given  to  isolated  determinations  than  to  determinations 
founded  on  lony  series  of  observations. 


Observations.              Year. 

Equation. 

Years. 

Weight. 

s. 

Bessel's  Uradley       .         1756 

.»•'  —  .  &4_r'  —  +  .06 

7 

3 

Picizzi                                   I  Soo 

.v'  —   .  2OJ''  =    —   .08 

(•> 

2 

V- 

Maskelyne     ...        1805 

•»•'  -  .i5/=  -  -'3 

2 

I 

Bessel  I    ....        iSiy 

y  -  .03,1-'  =  +  .05 

3 

2 

Bessel  II  .      .      .      .        1822 

.v'  +    .02/  —          .00 

5 

4 

Pond               .           .        1823 

x'  +  .03  v'  —  -f-  .07 

14 

A 

Struve        ....         1825 

x'  +  ,o$y  —  +  .04 

5 

4 

Argelander    .      .      .         1828 

x  +  .08  y  =  -t-  .06 

tf   '. 

•1 

Cambridge     .     ,     .        1835 

x'  +  .  isy  =  -  .03 

5 

2 

Edinburgh      ...        1838 

x'  +  .  isy  —  +  .04 

4 

2 

Airy      ,      1839 

x  +  .ig/  =   -  .05 

(, 

3 

Berlin                          .         1841 

i.v'  +  .  21  y  ~    —  .07 

4 

2 

Cambridge     ...         1841 

x'  -t-  .2i/  =  +  .06 

t 
5 

2 

Edinburgh     .      .      .        1842 

.»-'  +    .  22J-'  =     —    .01 

4 

2 

Pulkowa  ....         1845 

x  +  .25;''  —       .00 

8 

5 

Greenwich      .      .      .         1845 

x'  +   .25/  -~          .00 

6 

3 

Cambridge                .   !     1849 

x'  +   .2<)/  =  +   .06 

5 

2 

Greenwich      .      .      .         1851 

X    +   -3I/  —   +   .02 

6 

3 

Cambridge     .      .      .         1856 

.1-'  +  .  3&y  =  —  .01 

5 

2 

Greenwich      .                     1857 

x  +  .37  y  -  -  -oi 

7 

3 

Paris                      .      .         1859 

x'  -4-    10  v  —         .00 

10 

Greenwich     ...        1864 

i    '  j7  j 
x  +  .44,1''  =    -  .02 

•7 

3 

Washington  .      .      .         1864 

x  +  .447'  =  +  .05 

4 

4 

Paris                            .         1865 

x'  +  .45  y  =  \-  .  1  1 

2 

2 

Washington   ...        1867 

x  +  .47  y  -  +  .05 

4 

4 

Greenwich      .      .      .         1869 

1 

x'  +  .49y  =  -  .03 

3 

3 

Treating  these  equations  by  the  method  of  least  squares,  we  are  led  to  the  normal 

equations — 

74     y'  +  14.9  i/  —  +  i  ".02 

14.9  x' +    7.4  #' =  4-0*.  1 7 


the  solution  of  which  gives — 


x'  =  -f  o'.oio 
])'•=.  — 


5— F 


34 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 


The  common  correction  to  be  applied  to  the  star-positions  first  dednced  is  there 

fore  — 

+  o".oi6  —  .009  T 

T  being  the  fraction  of  a  century  after  1820.  Applying-  these  constant  corrections  to 
the  values  of  x  and  y  previously  found,  we  shall  have  a  set  of  corrections  to  the  Ta 
bulae  Regiomontanae  which  we  may  suppose  to  be  free  from  all  systematic  error  what 
ever,  at  least  so  far  as  it  is  possible  to  free  them  with  our  existing  data. 

§  6- 

CORRECTIONS    FOR    THE    TERMS    MULTIPLIED    BY    THE    SECOND    AND    THIRD 

POWERS  OF  THE  TIME. 

We  have  hitherto  proceeded  on  the  supposition  that  the  necessary  corrections  to 
the  Tabulae  Regiomontanse  increase  uniformly  with  the  time.  But,  on  computing  the 
co-efficients  of  the  second  and  third  powers  of  the  time,  in  the  expression  for  the  mean 
right  ascension  of  the  star  at  any  epoch,  and  comparing  them  with  those  found  by 
induction  from  the  mean  places  of  the  TabuLe  Regiomontanse,  it  is  found  that  this  sup 
position  is  not  correct.  In  fact,  it  appears  that  Bessel  computed  the  second  differential 
or  secular  variation  of  the  annual  variation  without  taking  any  account  of  the  effect  of 
proper  motion,  and  that  he  considered  it  constant,  thus  neglecting  the  third  derivative 
entirely. 

The  following  table  shows,  i,  the  secular  variation  of  the  precession  actually 
employed  in  the  Tabulae  Regiomontanse  ;  2,  the  adopted  secular  variation  of  the 
annual  motion  for  1800,  computed  from  the  co-efficients  of  Struve  and  Peters,  using 
Hill's  formulae  in  the  "Star  Tables  of  the  American  ephemeris,"  page  19;  3,  the 
resulting  correction  to  the  first  value;  4,  the  variation  of  the  secular  variation  in  one 

century,  or  the  value  of  —  pr^g—  >  tne  um'*  °^  r'on^  ascension  being  os.ooi,  and  the  unit 

Cl   X 

of  T  one  century;   5  and  6,  the  corrections  to  #  and  y  necessitated  by  the  addition  of 
these  terms  to  the  positions  of  the  Tabula?  Regiomontanse. 

These  last  terms  are  derived  as  follows:  Representing,  for  brevity,  by  a  and  b  the 

,  d*  R.  A.      ,  d3R.A.   .,       .  ,  .  .          , 

corrections  to  JJessel  s  assumed  values  ot  —  —  —  —  and  —  T-TT-PT—  .  the  right  ascensions  ot 

dl"  d  T3 

the  Tabulae  Reiomontanse  should  have  received  the  correction— 


for  the  epoch  of  each  catalogue  before  the  comparison  with  the  catalogue.  T  being 
here  counted  from  1  800,  the  values  of  Cj,  c.2,  etc.,  already  given,  would  have  been  thus 
changed  by  the  following  amounts: 

«?!    —  —  O.  lOrt  +  O.O2& 

c.3  to  c4  —      o. 

c5  —  —  .01  a 
c6  =-•  .03  a 
c,  —  —  .03  a 


S>                       T^T_-L          _ 

.03   ft 

.04  a 

.IO«  — 

.026 

?1=- 

.2Ort  — 

.066 

CORRECTIONS  FOR  TERMS  MULTIPLIED   BY  SECOND  AND  THIRD  POWERS  OF  TIME.         35 

Substituting  these  values  of  Ci,  etc.,  in  the  expressions  for  .rand  y,  (p.  10,)  we  find — 

—  —  .08  a  —  .o  i  6 
—  .  i3«  —  .066 
the  values  of  which  are  given  in  the  last  two  columns. 


Star. 

d?  R   A 
Bessel's  —  rv-~~ 

a  1  ' 

From  Struve, 
(1800.) 

h 

AJ:                    \y 

y     Pcgasi   .... 

+     97° 

+     953 

12            +       63 

O              —      2 

a     Arictis  .... 

4-   2010 

+   1998 

12     :     4-     41 

0                       I 

a     Ccti 

4-        O7O 

+            QCQ                              

'    T                                                       II 

! 

V> 

939 

1    1                              "J"                1_| 

4~       I               41 

a     Tauri      .... 

4-    1090 

4-    1065 

25          -     39 

4-2             -h      6 

ft    Orionis. 

+    430 

+      4°7 

23                  14 

4-2             +4 

,3    Tauri     .... 

4-      890 

4-      862 

28          -     95 

+     3     :       +     9 

a     Orionis. 

+      31° 

+  '  303 

7    ;         42 

+      i             +4 

a     Canis  Majoris. 

4-       40 

64              -    i 

Hi             ii 

a     Canis  Minoris 

440 

515 

75          ~     20 

ft    Gcminorum 

-    I22O 

-    1216             4- 

;                  64 

o            4-4 

a     Ilydrx  .... 

150 

157 

7     '•     +     25 

o           —     i 

«     Lconis  .... 

-     IO2O 

-    1039 

9+37 

4-1                    o 

(3     Lconis  .... 

770 

7&S            4- 

2    ;    4-    03 

I                      4 

a     Virginis 

4-    1  1  20 

4    1  1  1  r 

9     ;     +     5<-> 

0-2 

a     Bootis  .... 

4-      1  20 

4-       202                 4- 

)2          4-     44 

7              -    '3 

a-    Librae    .... 

+    1550 

+  1537 

3+17 

4-1             +1 

a     Coronas 

+      240 

4-     228 

2          4-     17 

4-1             4-1 

a     Serpcntis    . 

4-      630 

4-609           -     : 

!I              4-          5 

4-2               4-2 

o     Scorpii  .... 

+    1570 

+   1549                 : 

!I             —       56 

4-      6 

re     Hcrculis     . 

+      37° 

+     353                  i 

7     i             n 

+      1             +3 

1 

a     Ophiuchi     . 

+      35° 

+     344 

6                  13 

4-1               4-2 

a     Lyras      .... 

4-      1  60 

4-      108                    i 

a    |    -      6 

+      4             +7 

7     Aquila;  .... 

-       So 

99                    l 

9                    5 

4-      2      •        4-3 

«     Aquila:  .... 

140 

—      180 

\o                    1 

+      5 

ft    Aquilae  .... 

-      150 

140            4- 

0         —       8 

i                     I 

«2   Capricorn*  . 

810 

834                    • 

4                 IS 

4-     2     !        -4-4 

<z     Aquarii 

430 

444                    1 

4         +35 

4-      i                    o 

a     Pcgasi  .... 

+     530 

+     523 

7+56 

0               —       2 

«     Andromcdic     . 

+   1770 

+    177° 

0     1     4-     83 

1 

i            -     5 

The  values  of  x  and  y,  or  the  correction  to  the  right  ascension  and  annual  varia 
tion  of  the  TabuLe  Regiomontanae  for  the  epoch  1820,  may  now  be  found  by  adding 
to  the  values  of  those  quantities  already  found  (p.  i  7)  the  constants  for  correction  of 
equinox  -f-  1 6  and  —  9,  and  the  corrections  from  the  above  table.  After  doing  this  we 
have,  for  convenience  of  application,  transferred  the  epoch  to  1800.  Adding  the 
co-efficients  for  the  second  and  third  powers  of  the  time,  we  have  the  following  values 
of  the  corrections  in  thousandths  of  seconds  of  time  to  the  right  ascensions  of  the 
Tabulae  Regiomontanee,  where  we  put  T  —  fraction  of  century  after  1 800 : 


.  ,6  RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

v.7 

Corrections  to  Tabula  Rey'wmontance. 

y  Pegasi,  4  R.  A.  -  +  26  -  1 38T  -     6Ta  +  ioT3 

a.  Arietis,  3  +  T57  6  +7 

«  Ceti,  +33+8i  6  +2 

a  Tiuiri,  3+90  -12  6 

/?  Orionis,  +41+27  -12  2 

/?  Tauri,  +66—165  -14  -16 

a  Orionis  +24-10  4  7 

a  Canis  Majoris,  +26+64  -52  2       +  j  (Auwers) 

a  Canis  Minoris,  -27+127  -38  3       +  q  (Auwers) 

/?  Geminorum,  —20  +  231  +    2  -11 

a  Hydra,  -20  +  276  4  +4 

a  Leonis,  -29  +  216  -10  +    6 

/?  Leonis,  0+193  +    i  +10 

a  Virginis,  -16+67  4  +    8 

a  Bootis  7+99  +41  +7 

a2  Libra?,  +80—241  6  +3 

a  Coronae,  -12+152  6  +3 

a  Serpentis,  +  48 — J36  -  10  +    i 

a  Scorpii,  +41  —  131  -  i  o  9 

a  Herculis,  5+66  8  —    2 

a  Ophiuchi,  -27  +  35°  3  —    2 

a  Lyra,  +2+86  -  26  i 

y  Aquila?,  +69  —  1 94  -  i  o  i 

a  Aquilo?,  +7+35  ~  2O  J 

13  Aquila>,  +39—216+5  i 

a2  Capricorni,  6  —      6  -12  —    2 

a  Aquarii,  -34+165  7  +6 

«  Tega.si,  +    9+    18  4  +9 

a  Andromeda1,  -12  +  185  °  +  H 

We  have  here  added  the  stars  Sirius  and  Procyori  to  the  list  because  Anwers  has 
determined  their  irregular  proper  motions  with  sufficient  accuracy  to  allow  of  their 
being  used  as  standards  of  comparison  fora  long  time  to  come.  The  following  are  the 
details  of  the  operations  by  which  Auwers'  results  have  been  applied  in  the  above 
table. 

PKOCYON. 

By  his  researches  on  the  proper  motion  of  Procyoa,  Auwers  has  found  a  certain 
correction,  q,  to  its  right  ascension,  denned  as  follows:  Represent  by  f/{,  (/3,  etc.,  to  q6,  the 
several  corrections  to  be  applied  to  the  right  ascension  of  Procyon  in  Wolfers'  "Tabula; 
Reductionum,"  in  order  that  the  respective  differences  between  the  corrected  right 
ascension  and  the  right  ascensions  of  a  Tauri,  /3  Orionis,  a  Orionis,  /?  Geminorum,  a 
ITydrse,  and  a  Leonis,  given  in  the  same  tables,  may  be  correct.  Then  Auwers  puts— 


CORRECTIONS  FOR  TERMS  MULTIPLIED  BY  SECOND  AND  THIRD  POWERS  OF  TIME.        37 

It  follows  that  if  we  represent  by  ^,  J.,,  etc.,  to  Je,  the  definitive  corrections  to  Wolfers' 
positions  of  these  six  stars,  the  correction  to  Wolfers'  position  of  Procyon  will  be  — 

\24  +  q 

But  this  result  requires  a  small  change  arising  in  this  way.  Wolfers  and  Auwers  have 
assumed  that  the  corrections  to  the  mean  places  of  the  Tabula?  Regiomontana?  increase 
uniformly  with  the  time.  In  other  words,  it  is  assumed  by  Auwers  that  q  is  of  the 
form  — 

x  +  yt  +  P 

P  being  the  purely  periodic  terms  depending  on  the  orbital  motion.  But,  on  examin 
ing  the  values  of  the  terms  depending  on  the  second  and  third  powers  of  the  time  in 
the  preceding  table  it  will  be  seen  that  the  correction  is  really  of  the  form  — 

+  i  s  rti  1      i      T  ^ 

///  —  o".03i  I  -+  P 

To  determine  rigorously  the  correction  to  q  which  would  have  been  introduced  had 
Auwers  taken  account  of  the  term  in  Ts  Avould  be  difficult.  Owing  to  the  extreme 
minuteness  of  the  correction,  Ave  assume  that  the  effect  of  changing  the  term  would 
correspond  to  the  effect  of  a  similar  change  in  the  equations  used  in  this  paper,  or— 

^lx  —  J  R.  A.  (1820)  —  —  .08  a  —  +  os.oo5 
^y  —  —  .i3«  =  —  .i3X  —  62     =  + 
The  correction  to  Auwers  will  then  be— 


T  being  the  fraction  of  a  century  after  1  800. 

The  value  of  this  correction  will  be  only  —.006   in    1/55,  and  will  not  exceed 
o".oo3  during  the  period  1  800  to  1  860. 

We  therefore  conclude  that  the  correction  to  the  position  of  Procyon  given  in  the 
Tabula?  Reductionum  is  — 

^  +  3  +  8T-3iT*  +  '/ 

It  remains  to  determine  d.     Subtracting  Wolfers'  corrections  to  the  positions  of 
the  Tabula?  Regiomontanae,  as  found  in  the  Tabula?  Reductionum,  pp.  xxxv,  xxxvi, 
from  the  corrections   given  in  the  preceding  table,  it  seems  that  our  corrections  to 
Wolfers1  positions  are  as  follows  : 
a  Tauri 
ft  Orionis 
a  Orionis 
ft  Gerninorum 
a  Hvdra? 

*/ 

a   Leonis 

Mean  J-     -24  -  14  T  7  T«  3  T3 

The  correction  to  Wolfers'  position  of  Procyon  is,  therefore, 

-2i-6T-38T'-3T'+2 

Adding  Wolfers'  correction  to  Bessel's  position  of  Procyon,  which  is  — 

-6  +  I33T* 


—  25 

-  '  7  T 

-   I  2   ' 

r8  -  6  pr3 

+      2 

-61 

-   I  2 

—       2 

+       I 

—  99 

4 

Ij 

—  35 

+  '5 

+      2 

-  14 

-40 

+  28 

4 

+    4 

-47 

+  5' 

-  10 

+    6 

^8  RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

J 

we  have  the  result  given  in  the  table  for  the   correction  to  the  right  ascension  of 
Procyon  in  the  Tabulae  Regiomontanae. 

SIEIUS. 

Auwers'  correction  to  the  position  of  Sirius  is  given  in  a  form  similar  to  that 
adopted  for  Procyon,  but  the  stars  of  comparison  are  /?  Orionis,  a  Orionis,  and  Pro 
cyon.  The  positions  adopted  for  comparison  are  those  of  the  Tabulae  Regiomontanpe, 
except  that  the  right  ascension  of  Procyon  is  first  corrected  by  the  quantity— 

-os.025  —  os.ooi3<86  (t—  1825)  —  oa.o;o5  cos  9°.oo634  (t  —  179557) 
or  by  the  quantity  — 

+  10-  139  T  +  P 

P  being  the  periodic  term.     The  terms  in  the  correction  to  the  relative  motion  of  Sirius 
which  depend  on  the  second  and  third  powers  of  the  time  are  — 

.        -34T2+2T3 

and,  reasoning  as  in  the  case  of  Procyon,  we  conclude  that  Auwers'  q  should  be  cor 
rected  by  the  quantity— 


The  value  of  this  correction  for  different  dates  is  — 

s. 

i  760  —  .005 

1780  +.001 

1  800  +  .004 

1820  +-004 

1840  +.002 

1860  --003 

1880       .  -.010 

The  correction  to  the  right  ascension  of  Sirius  in  the  Tabulee  Regiomontanse  is, 
therefore, 


4  being  the  mean  correction  to  the  right  ascensions  of  /?  Orionis,  a  Orionis,  and  Pro 
cyon,  the  latter  star  being,  however,  first  corrected  by  the  expression  given  above. 

Comparing  this  expression  with  Auwers'  q,  (A.  N.  No.  1382,  p.  62,)  the  value  of 
which  is  — 

39  —  1  19  T  —  os.o7O5  cos  n  (t  —  T) 
we  see  that  the  preliminary  correction  to  the  position  of  Procyon  is  — 

q  —  29  —  20  T 


CORRECTIONS  TO  DIFFERENT  CATALOGUES  TO  REDUCE  THEM  TO  A  MEAN  SYSTEM. 


39 


Subtracting  this  correction  from  the  above  correction  to  Procyon,  taking  the  cor 
rections  to  /?  Orionis  and  a  Orionis  without  change,  and  representing  by  z/,,  J.,,  and 
J-i  the  individual  corrections  to  Auwers'  assumed  positions  of  the  three  stars  of  com 
parison,  we  have— 

^  =  +  41  +    27  T  -  i  2  T-  -  2  T:i 
-fo—  +  24-      10  4         -7 

^3=+    2  +  147        -38         -3 
,4  =  +  22  +     55T--l8Ta   -4T3 

Adding  this  to  the  expression  which  replaces  Auwers'  g,  we  have  the  result  given 
in  the  above  table. 

§7- 

CORRECTIONS    TO    DIFFERENT    CATALOGUES    TO    REDUCE    THEM    TO   A  MEAN 

SYSTEM. 

The  preceding  investigations  have  resulted  in  giving  us  a  set  of  29  right  ascen 
sions,  which  we  may  suppose  to  be  free  from  all  systematic  errors  whatever,  and  of 
which  we  may  suppose  the  mean  accidental  error  to  be  much  less  than  that  of  any  one 
catalogue  determination.  We  may,  therefore,  suppose  any  systematic  differences  which 
exhibit  themselves  between  the  right  ascensions  of  any  catalogue  and  those  just  obtained 
to  be  due  to  errors  of  the  former,  and  from  the  entire  body  of  the  differences  we  may 
obtain  the  correction  necessary  to  reduce  it  to  the  standard.  This  correction  may  be 
considered  to  consist  of  three  parts— 

1.  A  constant,  which  we  may  consider  as  a  correction  to  the  adopted  equinox. 

2.  Periodic  terms  depending  on  the  right  ascension,  which  in  all  recent  catalogues 
we  may  suppose  to  be  of  the  form— 

m  cos  a.  +  n  sin  a 

and  to  which  terms  of  double  period— 

in'  cos  2  a  +  n'  sin  2  a 

may  be  added,  if  it  be  deemed  necessary.  In  some  of  the  older  catalogues,  especially 
in  Bessel  I,  the  terms  of  double  period  appear  to  be  quite  sensible,  but  thev  ought  not 
to  be  real  in  any  modem  catalogue. 

3.  A  term  depending  on  the  declination,  which,  within  the  limits   in   which  the 
present  investigations  are  applicable,  may  be  regarded  as  proportional  to  the  simple 
declination.     In  application,  it  might  be  better  to  suppose  it  proportional  to  the  sine 
of  the  declination.     The  limits  in  question  may  be  considered  as  40°  north  declination 
and    30°  south  declination,  but  the  correction   in   question  will    necessarily  become 
uncertain  as  we  approach  either  limit. 

These  three  corrections  have  been  obtained  in  the  order  in  which  they  are  num 
bered,  as  follows:  Represent  by— 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 


the  several  excesses  of  the  standard  right  ascensions  over  those  of  the  catalogue  to  be 
compared.  The  mean  value  of  c  may  then  be  taken  as  the  correction  for  equinox, 
which  we  may  represent  by  e.  Subtract  e  from  each  individual  c,  and  represent  the 
residuals  by  c't,  e'2,  etc,  The  values  of  m  and  n  may  then  be  obtained  in  the  same 
manner  by  which  the  periodic  terms  of  the  residuals  of  the  separate  catalogues  have 
been  obtained.  But  the  addition  of  Sinus  and  Procyon  to  the  list  'of  stars  will  make 
a  slight  change  in  the  equations,  so  that  we  shall  have— 


i  1.9  in  —  2  c'  (cos  a  —  .0 


sn 


i  7.  i  n  —  2  c'  (sin  a  —  .04  cos  a) 
The  values  of  the  coefficients- 

cos  a  —  .03  sin  a  and  sin  a  —  .04  cos  a 

for  each  star  are  given  in  the  following  table. 


Co-eflicient  for  — 

Star.                           cos  «      .           sin  « 

n  .gin               1  7  .  1  72 

a 

Andromcdtc  .      .        +     I.oo                    .00 

+      i  .  oo                    .04 

7 

Pegasi                   .    '     +      i.oo          +        .03 

+      i.oo          —        .01 

a 

Arietis       ...         +        .87     ,     +        .50 

+        .86          +        .47 

a 

Ceti      ....    |     +        .73          +        .69 

+        .71          +        .66 

a 

Tauri   .....+        .39          +        .92 

+        ,36          +        .9° 

P 

Orionis      .      .      .    j     +        .22          +        .97 

+        .19          4-        -96 

ft 

Tauri   ....'+        .19          +        .98 

+        .16          +        .97 

n 

Orionis     ...         +        .06     !     +      i.oo 

+        .03          +     i  .  oo 

a 

Canis  Majoi  is      .                   .16          +        .99 

.19         +     i.oo 

a 

Canis  Minoris     .                  .38          +       .92 

.41          +        -93 

ft 

Geminorum    .      .    :                .40          -f-        .91 

-43          +        -93 

a 

Hydras      ...                   .76          +        .  64 

-        .78          +        .67 

a 

Leonis       ...                   .86          +        .51 

.88          +        .54 

(3 

Leonis       ...                i.oo         +       .09 

I  .  00             +          .13 

a 

Virginis    ...                   .95          —        .32 

.94                    .28 

a 

Bootis.      .      .      .                  .85          —       .53 

-     .83       -     .50 

a- 

Librae  ....                   .77                    .6} 

.75          -        .61 

a 

CoronacBorealis.                  .62                   .78 

-        .60                    .76 

a 

Serpentis  ...                   .59                    .81 

-        -57          -        -79 

a 

Scorpii       ...                   .43                    .90 

.40                    .88 

a 

Hcrculis   .      .      .                   .23                    .97 

.  20                    .  96 

a 

Opbiuchi  .                               .16                    .99 

.13                    .98 

a 

Lyric   .                          4-         n                     on 

+        .16                  I  .  oo 

7 

Aquilae                  .    j     +        .41                     .91 

4-        .44                    -93 

a 

Aquilse      .      .      .        +        .43                    .90 

+        .46                    ,92 

0 

Aquilae                  .    J     +        .45                    .89 

+        .48                    .91 

0« 

Capricorn!      .      .        4-        .53                    .85 

+          .56       :                     -37 

a 

Aquarii      ...         -fr        .86          —        .51 

+       .88     '              .54 

a 

Pegasi       ...         +        .96                    .28 

4-        .97                    -32 

CORRECTIONS  TO  DIFFERENT  CATALOGUES  TO  REDUCE  THEM  TO  A  MEAN  SYSTEM.        41 

Subtracting  from  each  c'  the  corresponding  value  of  — 

m  cos  a  -\-  n  sin  a 

the  residuals  ought  to  l>e  purely  accidental  errors,  unless  they  are  found  to  depend  on 
the  declination.  In  this  case,  if  we  suppose  a  term  proportional  to  the  simple  declina 
tion,  we  may  represent  it  in  the  form  — 

.Dec.—  10° 


the  zero  of  reference  being  10°  north  for  convenience,  because  that  is  nearly  the  mean 
declination  of  all  the  adopted  standard  stars,  and  the  degrees  of  declination  being 
divided  by  10  in  order  to  have  a  more  convenient  factor.  We  may  then  find  d  by 
solving  the  equations  of  condition  formed  by  putting  the  above  expression  equal  to  c" 
for  each  star.  The  following  is  a  convenient  and  sufficiently  accurate  expression  for  d: 

60  d  =1  -\-  2  c"  of  a  Andromedac. 

+  i  -  -    a  Arietis. 

—  i  .  .a  Ceti. 
+  I  .  -   or.  Tauri. 

—  2  .  .   /?  Orionis. 
+  2             /?  Tauri. 

—  2  .  .    a  Canis  Majoris. 
+  2  .  .   /?  Geminorum. 

—  2  .  .    a.  Hydra;. 
-)-  i  .  .   /?  Leonis. 

-  2  .  .    a  Virginis. 

+  I  .  -    a  Bootis. 

—  2  .  .or  Libra;. 
-f-  2  .  .    a.  Corona?. 

—  2  .  .    a  Scorpii. 
+  i  -  -    «  llerculis. 
-f-  2  .  .a  Lyrse. 

—  2  .  .or  Capricorni. 

—  i  .  .        Aquarii. 

As  an  example  of  the  method  we  may  discuss  the  twelve-year  series  of  Paris 
observations,  extending  from  1856  to  1867.  The  different  results  are  shown  in  the  fol- 
lowin<>-  table.  First  we  trive  the  mean  correction  to  Le  Vemer's  fundamental  cata,- 

O  «5 

logue  deduced  from  all  the  observations,  using  the  mean  results  given  in  each  volume. 
Next  we  have  the  resulting  corrections  to  Bessel.  Subtracting  the  correction  to  Bessel 
for  the  mean  epoch,  we  have  the  correction  to  the  standard  of  this  paper,  or  —  c.  The 
mean  value  of  c  is  found  to  be  32,  which  is  the  value  of  e.  Adding  this,  we  have  —  c'. 
6  —  F  s 


42  RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 

From  the  values  oi  c'  we  find — 

m  =  +  20 
n  =  —  23 

Adding  20  cos  a  —  23  sin  a,  we  have  c".     From  c"  we  have — 

f?-  +  9 

Applying-  9  -  to  —  c"  we  have  r,  or  the  definitive  corrections  to  the  standard 

given  by  the  Paris  observations. 


Star. 

Corrections  to  — 

-  c 

Computed 
periodic 

terms. 

—  c" 

r 

LeVerrier. 

Tab.  Reg. 

a    Andromeda:  . 

-     15 

+     57 

•     47 

—     15 

+      20 

+       5 

+       21 

7     Pcgasi 

-      16 

—    '27 

-     69 

—     37 

+      20 

-     i? 

-      13 

a     Arietis       ...                31 

+     33     ;     -     57 

-     25 

+       5 

—      20 

—       8 

a     Ccti      .... 

+     34 

+     53     i     -     27 

+       5 

i 

+       4 

—         2 

n     Tauri   .... 

—     25 

+     49 

+       3 

+     35 

13 

+      22 

+      27 

/?    Orionis 

+     52 

+     76 

+     23 

+     55 

—     18 

+     37 

+       21 

,3    Tauri   .      .      .      . 
a     Orionis 

-       6 
+     16 

-     59 
+     13 

-      18 
3 

+     M 
+     29 

19 

—      22 

5 
+      7 

+       II 

+       4 

a     Canis  Majoris 

(-   228)         +39                    4 

+     28 

—      26 

+         2 

—       22 

a     Canis  Minoris     . 

(+     SO 

+     37 

4 

+     28 

-      29 

—         I 

—       6 

ft    Geminorum   . 

-       6 

+     92 

24 

+       8 

-      29 

—      21 

-       5 

a     Hydra; 

+     61 

+   157 

+       12 

+     44 

-     3° 

+       M 

-         2 

a     Lconis 

+      20 

+     98                  o 

+     32 

-    29 

+       3 

+       6 

ft    Leonis 

+       4 

+     86         -     32 

o 

—      22 

—      22 

-    17 

a     Virginis    . 

+     28 

o 

24 

+       8 

II 

-       3 

—      21 

o     liootis       .      .      , 

+       7 

+     30 

-     39 

7 

—       4 

—       II 

—         2 

a2   Librx  .... 

+     23 

—     66 

+       I 

+     33 

0 

+     33 

+       10 

n     Corona 

—     25 

+      IS 

—     60 

-     28 

+       6 

—       22 

7 

n     Serpcntis  . 

+     28 

74                  36 

4 

+       7 

+       3 

o 

a     Scorpii 

+     46 

-     36          +6 

+     38 

+       12 

+     50 

+     18 

n     Ilcrculis   . 

—     ii 

27 

-     59 

27 

+     IS 

-       9 

4 

a     Ophiuchi  . 

I 

+   146 

37 

—       5 

+     20 

+     15 

+     IS 

ft     Lyne    .... 

-     63 

•     48             -93 

-     61 

+    25 

-     36 

—       10 

7     Aquilic 

-       5 

-   "3 

-     03 

-     3i 

+    29 

—         2 

—         2 

a     Aquilx      . 

—     24 

—     44 

-     65 

—     33 

+    29 

-       4 

-       5 

ft    Aquilse 

+       6 

-   148 

—     60 

-     28 

+    29 

+        i 

3 

'i2    Capricorni 

+     36 

•     49 

-     33 

—       I 

+   30 

+     29 

+       8 

a     Aquarii 

+     10 

+       9 

-     55 

-     23 

+    29 

+       6 

4 

n     Pegasi 

-     42 

42 

—     62 

-     30 

+    25 

"       5 

1 

The  systematic  corrections  obtained  in  this  way  for  a  number  of  catalogues  are 
given  in  the  following  list.  The  year  attached  to  each  catalogue  is  that  to  which  it  is 
reduced.  Systematic  corrections  much  smaller  than  the  accidental  errorrs  of  the  cata 
logue  may  generally  be  omitted  entirely. 


CORRECTIONS1  TO  DIFFERENT  CATALOGUES  TO  REDUCE  THEM  TO  A  MEAN  SYSTEM.     43 

SYSTEMATIC  CORRECTIONS  TO  BE  APPLIED  TO  THE  RIGHT  ASCENSIONS  OF  EQUATORIAL  AND  ZODI 
ACAL  STARS  IN  DIFFERENT  CATALOGUES  TO  REDUCE  THEM  TO  A  MEAN  SYSTEM. 

AUWERS'  BRADLEY,  (1755.)* 
4  rr — os.o  I  o  +  OS.OO3  cos  a  —  os.OO7  sin  a 

The  mean  value  of  the  periodic  terms  being-  less  than  one-third  the  probable  error 
of  the  individual  star-places,  they  may  be  neglected  entirely. 


BESSEL'S  FUNDAMENTA,  1755. 
J  zr  —  oa.oi 


DE  ZACII'S  FIXARUM  PRECIPUARUM  CATALOGUS  Novus  EX  OHSERVATIONIBUS  ASTRONO- 

3IICIS    IN    SPECULA    ASTRONOMICA   GoTHANA    ANNIS    1787,    1788,    1789,    1790,  HABITIS 
ET    AD    INITIUM    ANNI    MDCCC.    CONSTRUCTS. 

Dec.  —  i  o° 

4  zz:  +  o".36  —  os.03  - 

10° 


PlA/ZI,    l8oO. 

,  Dec.  —  10° 

=  +  o".io-foa.02S          — o— 


MASKELYNE,  (1805.) 

Catalogue  in  vol.  4  of  Ms  observations,  second  part,  p.  1 12. 

Dec.  —  10° 

—  -|-  o  .  145  -|-o  .019  cos  tx  -f-  o  .006  sin  a  —  o  .012  - 


AUWERS'  CACCIOTORE,  (1805.) 
zz  —  O8.oo6  —  OB.O  10  cos  a  —  O8.O33  sin  a 


BESSEL'S. FIRST  FUNDAMENTAL  CATALOGUE,  (1815.) 
^  —  —  o".O3  7  —  os.O3  7  cos  a  -f-  o".O3  7  sin  a  . 


BESSEL'S  SECOND  FUNDAMENTAL  CATALOGUE,  (1825.) 
^  —  -f  o".O2o  —  o".oo6  cos  a  -\-  o".oio  sin  a 


STEUVE,  (1825.) 

,  Dec.  —  10° 

zr  — O  .O2O  -f  OH.OO7  cos  a  -{-  o  .005  sin  a  -f  o  .008  -       — o~ 


'AuworH1  catalogue  being  Htill  iiiipul)li.Hli('il,  tliin  correction   must  lie  uiiiU'i.stood   IIM  ap|iliculilu  only  to  tho  [iri1- 
limiuury  catalogue,  in  which  no  correction  to  BcHM-l's  equinox  has  liecn  ili'tcrinini-U  or 


44  RIGHT  ASCENSIONS  OF  FUNDAMENTAL  STARS. 

BRINKLEY,  (1824.) 


ARGELANDER,  (1830.) 

,-  Dec.  —  10° 

—  +  os.oi  7  —  o  .006  - 


10 


G-ILLISS'S  FIRST  WASHINGTON  CATALOGUE,   1840. 

{From  observations  made  at  the  old  Naval  Observatory  during  the  years  i838-'42.     Wash 
ington,  1 846.  .  8vo.) 
A  —  -f-  O8.O3  2  cos  a  —  os.O28  sin  a. 


POND,  1830. 

J  — —  os.o56  —  os.oi5  sin  a. 

(For  stars  in  south  declination  the  additional  positive  correction  O8.ooi5  X  south 
declination  in  degrees  is  strongly  indicated.) 


AIRY'S  CAMBRIDGE  CATALOGUE. 
In  the  Memoirs  of  the  Royal  Astronomical  Society,  vol.  XI. 

4  —  —  os.O24  —  os.oi8  sin  2  a 

(The  term  of  double  period,  though  a  priori  somewhat  improbable,  is  strongly 
indicated  by  the  comparisons. 


TAYLOR,  (1835.) 

General  catalogue  of  the  principal  fixed  stars  from  observations  made  at  the  honorable  the 
East  India  Company's  observatory  at  Madras  in  the  years  1830-1843. 

Dec.  —  10° 
^/  —  —  O8.O47  +  o  .023  cos  a.  -(-  o  .001 1  —     — • 

The  last  term  is  far  within  the  probable  error  of  the  catalogue-positions. 


GREENWICH  12-YEAR  CATALOGUE. 

Catalogue,  for  1840. 
=  +  os.o67  -f  os.oi4  cos  a  —  01034  sin  a 

Catalogue  for  1845. 
—  -|-  o".oi9  -|-o8.oo7  cos  a  —  os.o25  sin  a 


CORRECTIONS  TO  DIFFERENT  CATALOGUES  TO  REDUCE  THEM  TO  A  MEAN  SYSTEM.     45 
ROBINSON'S  ARMAGH  CATALOGUE,  (1840.) 

4  r=  -f  O".O2O. 

(No  positions  of  fundamental  stars  are  given  in  this  catalogue.  The  correction 
has,  therefore,  been  derived  from  a  comparison  of  all  the  stars  south  of  45°  north  polar 
distance  which  are  common  to  Armagh  and  the  Greenwich  6- Year  Catalogue  for 
1 840,  and  observed  at  least  five  times  at  each  observatory.  The  differences  between 
these  catalogues  were  corrected  for  the  error  of  the  Greenwich  catalogue,  and  the  out 
standing  differences  were  presumed  to  be  due  to  error  of  the  Armagh  catalogue.) 


PULKOWA,  (1845.) 
4  =  -f  o". 


RADCLIFFE  CATALOGUE,  1845. 
4  zz  —  os.oo9  -f-  o".O47  cos  a  —  os.O2o  sin  a 
(The  positions  of  the  fundamental  stars  in  this  catalogue  give  the  correction  — 

z/  —  +  os.ooi  +  C/.O55  cos  a  —  o".O2 1  sin  a 

In  the  introduction  to  the  catalogue,  Mr.  Main  gives  a  comparison  of  this  catalogue 
with  Greenwich,  1845,  from  which  I  derive — 

Greenwich  —  Radcliffe  =  —  o".O3S  +  01032  cos  a  -(-  o".oo5  sin  a 
Adding  the  Greenwich  correction,  this  will  give,  for  Radcliffe, 

4  •=.  —  o".oi9  +  os.O39  cos  a.  —  o".O2O  sin  a 
The  mean  result  has  been  adopted.) 


GREENWICH  6- YEAR  CATALOGUE,  (1850.) 
l  —  —  o".oo3  -f-  o".oi  i  cos  a  —  o".O23  sin  a 


GREENWICH  7- YEAR  CATALOGUE,  (1860.) 
J  —  -f  o8.oi4  -f-  os.oi4  cos  a  —  o".oi2  sin  a 


GREENWICH  7-YEAR  CATALOGUE,  (1864.) 
d  —  +  o".O33  -|-os-QIo  cos  a.  —  oa.oio  sin  a 


PARIS  ANNUAL  CATALOGUES,  i856-'67. 

Dec.  —  10° 

-)-  oa.O32  -(-  os.O2o  cos  a  —  o  .023  sin  a  -f  o  .009  -        r~c— 


WASHINGTON  "POSITIONS  OF  FUNDAMENTAL  STARS,"  (1870.) 
(In  Appendix  III  to  the  Washington  Observations  fur  1867.) 


46 


RIGHT  ASCENSIONS  OF  FUNDAMENTAL  STARS. 


WASHINGTON  ANNUAL  RESULTS  OF  OBSERVATIONS  WITH  THE  OLD  TRANSIT-INSTRUMENT, 

(i862-'65.) 
^/  -)-  —  os.oo6  -f-  os.O2 1  cos  a  —  os.O24  sin  a. 


WASHINGTON  ANNUAL  RESULTS  OF  OBSERVATIONS  WITH  THE  TRANSIT  CIRCLE,  (1866-72.) 

A  —  -f-  os.oo6 

Those  of  the  above  corrections  which  form  a  considerable  part  of  the  probable  error 
of  the  catalogue-positions  are  tabulated  in  the  following-  tables.  In  the  corrections 
depending  on  the  declinations  the  constant  term  is  included,  except  for  Paris,  where  it 
is  included  in  the  first  table. 

Corrections  to  different  catalogues  depending  on  the  right  ascension. 


R.  A. 

Airy's 
Cambridge 
catalogue. 

Taylor's 
Madras 
catalogue. 

AIRY'S  GREENWICH  CATALOGUES. 

V 

Paris  annual 
catalogues, 
1855-67. 

'Radclifle 
catalo   ue, 
1845. 

1840. 

1845. 

1850, 

1860. 

1864. 

h. 

s. 

j. 

s. 

s. 

S. 

s. 

s. 

s. 

s. 

o 

—   .02 

—     .02 

+  .081 

+  .026 

+  .008 

+  .028 

+  .043 

+    .052 

+  .04 

i 

-   -03 

—     .02 

+  -072 

+  .020 

+  .002 

+  .025 

+  .040 

+    -045 

+  .03 

2 

-   -04 

-     -03 

+  .062 

+  .013 

—  .005 

+  .O2I 

-T--037 

+    .037 

+     .02 

3 

-   .04 

-     -03 

+  .053 

+  .OO6 

—  .Oil 

+  .015 

+  -033 

+    .030 

+     .01 

4 

-   -04 

-     .04 

+  .045 

+  .001 

-.017 

+  .OIO 

+  .029 

+     .022 

-     .01 

5 

-   -03 

-     .04 

+  .038 

—  .003 

—  .022 

+  .005 

+  .026 

+     .015 

—     .02 

6 

—    .02 

-     -05 

+  -033 

—  .006 

—  .026 

+  .OO2 

+  .023 

+     .OOg 

—     -03 

7 

—    .02 

-     .05 

+  .030 

—  .007 

—  .028 

.O 

+  .021 

+     .005 

-     .04 

8 

—     .01 

-  .66 

+  .031 

—  .007 

—  .028 

—  .003 

+  .OIg 

+     .002 

—     -05 

9 

—     .01. 

-  .06 

+  .033 

—  .004 

-.027 

—  .004 

+  .Oig 

+    .002 

—     .06 

10 

—     .01 

-  .07 

+  .038 

+  .OOI 

—  .024 

—  .004 

+  .0lg 

+    .003 

-     .06 

II 

—     .02 

-  .07 

+  .044 

+  .006 

—  .020 

—  .OO3 

+  .021 

+    .007 

-     .06 

12 

—     .02 

-  .07 

+  •053 

+  .OI2 

—  .OI4 

.0 

+  .O23 

+    .012 

-     .06 

13 

-     -03 

-  .07 

+  .062 

+  .018 

—  .008 

+  .003 

+  .026 

+    .Dig 

—     .05 

14 

—     .04 

-  -07 

+  .072 

+  .025 

—  .OOI 

+  .007 

+  .O29 

+    -027 

-     .04 

15 

-     .04 

-  .06 

+  .081 

+  .O32 

+  .OO5 

+  .013 

"+.033 

+    .°34 

-     -03 

16 

-     .04 

—  .06 

+  .089 

+  •037 

+  .016 

+  .018 

+  .°37 

+    -042 

—     .02 

17 

-     -03 

—  -05 

+  .096 

+  .041- 

+  .OI6 

+  .023 

+  .040 

+    .049 

.00 

"  18 

—    .02 

-  .05 

+  .IOI 

+  .044 

+  .02O 

+  .026 

+  .043 

+    .055 

+    .01 

19 

—    .02 

-  .04 

+  .104 

+  .045 

+  .O22 

+  .028 

+  .045 

+    -059 

+    .02 

20 

—    .01 

-  .04 

+  .103 

+  .045 

+  .O22 

+  .031 

+  .047 

+    .062 

+  .03 

21 

—    .OI 

-  .03 

+  .IOI 

+  .042 

+  .O2I 

+  .032 

+  .047 

+    .062 

+  .04 

22 

—    .OI 

-  .03 

+  .  096 

+  '°37 

+  .018 

+  .032 

+  .047 

+    .061 

+   .04 

23 

—    .02 

—    .02 

+  .090 

+  .032 

+  .014 

+  .031 

+  .045 

+    .057 

+  .04 

24 

—    .02 

—   .02 

+  .081 

+  .O26 

+  .008 

+  .028 

+  .043 

+    .052 

+  .04 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 
Come/ions  depending  on  the  declination. 


47 


Dec. 

N.  P.  D. 

Piazzi, 
1800. 

Struve, 
Posit.  Med. 

Argelander, 
1830. 

Paris  annual 

catalogues, 
i856-'63. 

« 

s. 

s. 

s. 

s. 

-t-40 

5° 

+    .18 

+   .'oo 

-     .001 

+  .-027 

35 

H 

+    -17 

.000 

+    .002 

+    .023 

30 

60 

+    -1C 

-   .004 

4-   .005 

+    .018 

25 

65 

+    -14 

-  .008 

+   .008 

+    .014 

20 

7" 

+    -13 

—   .012 

+     .Oil 

+    .009 

15 

75 

+    .11 

—    .016 

+   .014 

+    .004 

IO 

So 

+    .10 

—     .020 

+   .017 

.0 

+  5 

§5 

+   .09 

-     .024 

+    .020 

-   .004 

o 

90 

+   .07 

—     .028 

+   -023 

—   .009 

—  5 

95 

+   .06 

—     .032 

+     .021 

-   .014 

—  10 

100 

4-   .04 

—     -036 

+     .029 

—   .018 

-15 

105 

+   .03 

-    .040 

+     .032 

—     .022 

—20 

no 

+    .02 

-    -"44 

+     -035 

-     .027 

-25 

»5 

.OO 

-    .048 

+     .038 

—     .032 

§8.. 
DEFINITIVE  BIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Applying  these  corrections  to  the  catalogue-positions,  we  may  deduce  definitive 
corrections  to  the  individual  right  ascensions  of  the  stars  under  consideration.  If,  in 
doing  this,  we  employed  only  the  catalogues  used  in  the  first  determination,  and 
assigned  the  same  relative  weights  as  in  the  first  discussion,  we  should  reproduce  the 
right  ascensions  already  found,  lint  the  weights  were  first  assigned  solely  with  respect 
to  the  probable  freedom  of  the  catalogue  from  periodic  errors  dependent  on  the  right 
ascension,  while,  after  elimination  of  these  errors,  they  should  be  assigned  with  respect 
to  the  purely  accidental  errors  of  the  determinations,  and  this  change  of  weights  Avill 
make  a  small  change  in  the  individual  definitive  positions.  We  can  also  add  a  number 
of  catalogues  not  employed  in  the  preliminary  determination,  particularly  the  Green 
wich  and  Paris  catalogues,  which  will  add  considerably  to  the  precision  of  the  results. 

The  details  of  the  process  are  shown  in  the  following  tables.  Under  each  star  is 
given,  in  the  first  column,  the  difference  of  excess  of  catalogue-place,  corrected  for  sys 
tematic  error  over  the  standard  place  already  deduced. 

In  two  cases  several  catalogues  of  small  weight  have  been  combined,  as  the 
exhibition  of  the  individual  results  of  each  catalogue  would  not  be  of  interest.  The 
first  set  are  the  catalogues  of — 

1'iazzi 1800,  weight  i 

Maskelyne        ....  1805,  weight  2 

Auwers'  Cacciotore     .      .  1805,  weight  2 

Bessel T8i5,  weight  i 

which  have  been  combined  into  a  single  result  for  the  mean  date   1807.     The  mean 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 


systematic  correction  was  so  small  that  only  the  correction  for  equinox  was  actually 
applied.  The  next  case  was  Airy's  Cambridge  catalogue  and  Pond's  general  catalogue, 
which  were  combined  into  a  single  result. 

No  corrections  depending  on  the  declination  have  been  applied,  except  in  the  case 
of  Paris  and  Argelander. 

Under  the  titles  Greenwich,  1869,  and  Washington,  1869,  are  given  the  mean 
results  of  the  Greenwich  and  Washington  observations  for  the  years  1868,  1869,  and 
1870,  without  any  correction  for  periodic  error. 

The  weights  assigned  are  constant  for  each  catalogue,  except  in  the  case  of 
Bradley,  where  the  weight  2  was  assigned  to  results  depending  on  more  than  100 
observations,  and  i  to  those  depending  on  less.  The  adopted  weights  and  general 
solution  are  as  follows,  the  time  T  being  reckoned  from  1845  : 


No. 

Catalogue. 

Date. 

Weight. 

T' 

Cx 

C    x 

C', 

C"y 

I 

Bradley    

1755 

i  or  2 

-    .90 

.035 

.054 

-  .46 

-    -63 

2 

Combined  catalogue     . 

1807 

I 

-    -33 

.029 

.024 

—    .20 

-    .14 

-3 

Bcssel  II        .... 

1823 

2 

—    .22 

.055 

.048 

-   .24 

-    .16 

o 

A 

Struvc  

l82( 

I 

—    .21 

.027 

.023 

—   .11 

—   -°7 

*f 

5 

Argelander    .... 

1828 

4 

—    -17 

.108 

.°95 

-    -39 

-   -24 

6 

Pond-  and  Airy    . 

1830 

i 

-    -15 

.026 

.023 

-    .08 

-   -05 

7 

Greenwich     .... 

1839 

2 

-    .06 

.050 

.048 

-    .08 

-   .05 

8 

Greenwich     .... 

1845 

2 

.O 

.049 

.048 

—    .02 

.0 

l845 

4 

.O 

.098 

.095 

—    .05 

J 

.0 

10 

Greenwich     .... 

1851 

3 

+    .06 

.072 

.071 

+   .05 

+   .06 

it 

Greenwich     .... 

1857 

3 

+    .12 

.070 

.071 

+   .14 

+    .12 

12 

1862 

4 

+    -17 

.092 

.095 

+    -25 

+     -21 

13 

Greenwich     .... 

1864 

4 

+    .19 

.090 

.095 

+   -33 

+     .26 

14 

Washington  .... 

1865 

5 

+    .20 

.112 

.118 

+    -44 

+   .35 

IS 

Greenwich     .... 

1869 

2 

+    -24 

.044 

.047 

+    .21 

+    -17 

16 

Washington  .... 

1869 

2 

+    .24 

.044 

.047 

+    .21 

+   -17 

The  equations  given  by  the  several  catalogues,  when  solved  by  least  squares,  lead 
to  the  result  that — 

4  R.  A.  for  1 845  =  2  Cx  X  >' 
J  Ann.  variation  —  2  Cy  X  r 

r  representing  the  correction  given  by  any  catalogue,  and  Cx,  Cy,  factors  constant  for 
each  catalogue,  except  that  there  will  be  two  systems  of  these  factors,  the  one  corre 
sponding  to  the  stars  for  which  the  results  of  Bradley's  observations  receive  the  weight 
i,  the  other  to  those  in  which  they  receive  the  weight  2.  The  former  are  designated 
by  one  accent,  the  latter  by  two.  In  the  case  of  Cx  the  differences  are  so  minute  that  a 
single  mean  value  has  been  actually  used. 

In  the  column  /,  under  each  star,  is  given  the  outstanding  residual  after  subtract 
ing  the  concluded  correction — 

4  x  +  T  4  y 

Sinus  and  Procyon  are  included  only  for  the  sake  of  comparison.     No  correction 
to  Auwers'  places  is  indicated. 


DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 
Residual  corrections  given  by  catalogues. 


49 


a  Androm. 

}-  Pegasi. 

a  Arietis. 

a  Ceti. 

a  Tauri. 

(i  Orionis. 

No. 

Cutnlou'iic' 

Year 

r 

/ 

r 

r 

r 

r 

r 

r 

r 

r 

r 

t> 

i 

Bradley       .... 

1755 

+  15 

+   12 

-  29 

-  49 

+    9 

+    4 

+  32 

+  45 

+  24 

+  17 

+    3 

+    6 

2 

Combined  catalogue 

1807 

+    9 

+    4 

+  23 

+  17 

—   12 

—  II 

—  24 

—  20 

—     G 

-  IS 

-    7 

-    4 

3 

Bessel  II    .... 

1823 

—   12 

-  IS 

+  27 

+  25 

—   12 

—    10 

-  30 

-  29 

+    5 

-    8 

-  25 

—  22 

4 

1824 

-  Gi 

-67 

-     i 

-    3 

+     2 

+    5 

+     I 

+    2-15 

-  28 

+  53 

+  56 

5 

Argelander 

1828 

+    i 

-    5 

+  15 

+  M 

-    8 

-    5 

—    10 

—    10 

+      2 

—  ii 

—   IO 

-     7 

6 

Pond  and  Airy 

1830 

+  27 

+  20 

+  40 

+  40 

+      2 

+    5 

-  28 

-  28 

+   22 

+    8 

—  ii 

—    8 

7 

Greenwich. 

1839 

+  26 

+   19 

o 

+      2 

28 

-  24 

+  4° 

+  38 

+  42 

+  28 

-  13 

—   IO 

8 

Greenwich  .... 

1845 

+    6 

—      I 

-  M 

—   II 

+  23 

4-  28 

-  23 

—  2G 

+  44 

+  29 

-    3 

o 

9 

1845 

+    8 

+      I 

-    3 

0 

+   M 

+  19 

+    4 

4-    I 

—    10 

-  25 

+    6 

+    9 

10 

Greenwich  .... 

1851 

+  23 

+  15 

-    7 

-      2 

0 

+    6 

o 

-    4 

+  21 

+    6 

—      2 

o 

ii 

Greenwich  .... 

1857 

+  28 

+  20 

-     2 

+    4 

-     9 

3 

-  M 

-  19 

+  IS 

+     2 

-  26 

-  24 

12 

Paris     

1862 

+  21 

+  J3 

—  '   13 

_     „ 

—     8 

—  .    i 

2 

4-  27 

+  11 

-4-    2T 

+   21 

'3 

Greenwich. 

1864 

+    9 

+    I 

-  18 

—    10 

-    6 

+     i 

+    8 

7 
+     2 

i     *•  l 
+   19 

+    3 

T^    *  * 

-    4 

•^J 

—      2 

'4 

Washington     . 

1865 

+     i 

-    7 

-  13 

-    4 

-    7 

o 

+  21 

+   15 

+   M 

-    3 

-    4 

—      2 

15 

Greenwich. 

1869 

+     2 

-    G 

—  21 

—  II 

-  30 

-  23 

—   IO 

-  16 

+  26 

+     9 

-  M 

—  u 

16 

Washington     . 

1869 

—  22 

-  30 

+    9 

—  10 

—        T 

+  43 

+  37 

+  II 

-     6 

+    6 

+    8 

No. 

Catalogue. 

Year. 

/3  Tauri  . 

o  Orionis. 

a  CanisMaj. 

a  CanisMin. 

/?  Gcmin. 

a  Hydra. 

I 

Bradley       .... 

1755 

+    4 

+     2 

-    9 

-    9 

o 

o 

. 

+  ii 

+    7 

-    5 

—    12 

2 

Combined  catalogue 

1807 

+  32 

+  25 

+  27 

+  26 

-    4 

-    6 

—     2 

-    4 

• 

Bessel  II    .... 

1823 

-    4 

—   12 

+      2 

o 

. 

. 

—   12 

-  13 

-  23 

-  24 

4 

Struvc   .            ... 

1824 

—  22 

-  30 

-   14 

-  16 

. 

. 

-  23 

-  24 

+  25 

+  24 

5 

Argelander 

1828 

+    3 

—    G 

+  M 

+   12 

-  17 

. 

+    S 

-     i 

4-  10 

+   10 

6 

Pond  and  Airy 

1830 

+  15 

+    G 

+  32 

+  3° 

+    5 

+  M 

+   10 

+    9 

+  48 

4-48 

7 

Greenwich  .... 

1839 

-    4 

*-  M 

+    4 

+     2 

. 

+      2 

-  24 

+   II 

+  ii 

-  15 

-  14 

8 

Greenwich  .... 

1845 

+    6 

-    4 

-    5 

-    7 

—    6 

. 

+  13 

+    3 

+    3 

+     3 

+    4 

Pulkowa    .... 

1845 

+  14 

+    4 

—  13 

—  15 

+    6 

+    6 

-+•         5 

+    6 

10 

Greenwich. 

1851 

+  17 

+    6 

-    S 

. 

+    4 

+      2 

-    3 

-     3 

T^         D 

-  6 

-    4 

ii 

Greenwich.     . 

1857 

+  34 

+  23 

+     3 

o 

+   12 

—   12 

+    7 

+    7 

—    4 

-    i 

12 

1862 

+  ii 

—    i 

+    4 

4-    i 

—   1  1 

—    6 

, 

—     c 

—     2 

-f-     i 

13 

Greenwich  .... 

1864 

+  16 

+    4 

+  16 

+  13 

* 

-   17 

. 

+    7 

D 

+    G 

D 

+    7 

-    S 

r 

'4 

Washington    . 

1865 

-  M 

-  2G 

+    i 

—      2 

. 

+   If) 

o 

—  ii 

—  IO 

—   12 

-    9 

15 

Greenwich  .... 

1869 

+  36 

+  24 

+    2 

—      I 

. 

-    3 

+  23 

+    9 

+   10 

4-  26 

+  30 

16 

Washington     . 

1869 

+  19 

+    7 

+    i 

-      2 

• 

+     I 

+  15 

-    7 

-    6 

-  23 

-  '9 

1 F  S 


RIGHT  ASCENSIONS  OF   FUNDAMENTAL   STARS. 

Residual  corrections  given  by  catalogues. 


a  Leonis. 

ft  Leonis. 

a  Virginis. 

a  Bootis. 

a2  Librae. 

a  Coronae. 

No. 

Catalogue. 

Year. 

r 

r 

r 

r 

r 

r 

r 

1 

r 

r 

r 

r 

i 

Bradley       .... 

1755 

+   12 

+    G! 

+    i 

+  ii 

-    5 

-  13 

—     I 

-    6 

-    7 

-    8 

+    5 

—  II 

2 

Combined  catalogue 

1807 

o 

—    8 

+  10 

+  17 

—      2 

o 

+    5 

+    3 

+    3 

—    i 

-  28 

-  35 

3 

Bessel  II    .... 

1823 

+      I 

-     5 

-  26 

—  20 

+  22 

+  27 

-    3 

-    3 

-    3 

-    8 

—    i 

-    5 

4 

Struve 

1824 

-    4 

—  10 

—   12 

«7 

-  16 

—    10 

-  17 

-  17 

-  19 

-  24 

+  30 

+  26 

5 

Argelandcr 

1828 

—    S 

-  14 

—      2 

+     3 

o 

+    6 

-    9 

-    9 

+  19 

+  13 

+    9 

+     6 

6 

Pond  and  Airy 

1830 

—    9 

-  15 

-    3 

+      2 

-  15 

-    8 

-    6 

-    C 

+  26 

+  20 

+  16 

+  13 

7 

Greenwich  .... 

1839 

+  3S 

+  32 

-    10 

-     6 

-  15 

—    C 

+  24 

+  25 

+  28 

+  23 

-    5 

-     6 

8 

Greenwich  .... 

1845 

+  ii 

+     5 

-  30 

-  26 

-  15 

-     5 

+    6 

+    7 

+  30 

+  24 

+    9 

+    9 

Pulkowa     .... 

1845 

+      2 

-    4 

+     G' 

+    10 

+    8 

+  18 

+  10 

+  ii 

—  ii 

—  17 

—      2 

10 

Greenwich  .... 

1851 

+     6 

+     i 

+     4 

+    S 

-  16 

+  15 

+  iG 

+    4 

J  / 
—      2 

+  28 

+  28 

ii 

Greenwich  .... 

1857 

+  16 

+  ii 

I 

+      2 

-36 

-  24 

+     8 

+  10 

-     9 

-   15 

-t-  15 

+  17 

12 

Paris      

1862 

+     f> 

+    i 

-  17 

-  '4 

—  21 

-     8 

—      2 

o 

+  10 

+    3 

-    7 

-     5 

13 

Greenwich  .... 

1864 

—     5 

-    10 

+    5 

+    8 

—  22 

-    9 

-    6 

4 

+  16 

+    9 

0 

+    .3 

M 

Washington     . 

1865 

+     5 

0 

-     3 

0 

+      I 

+  15 

—   12 

—  10 

0 

-    7 

-  18 

-  15 

15 

Greenwich  .... 

1869 

+  ii 

+    & 

+   12 

+  14 

—   21 

-    7 

-   19 

—  16 

+   10 

+    3 

—  10 

-    6 

16 

Washington 

1869 

+     6 

+     i 

-    5 

-    3 

~  14 

0 

—    10 

-    7 

+   20 

+  13 

-  25 

—  21 

No. 

Catalogue. 

Year. 

a  Serpentis. 

a  Scorpii. 

a  Herculis. 

a  Ophiuchi. 

a  Lyrx. 

y  Acjuiloe. 

i 

Bradley       .... 

1755 

+     5 

+     i 

-  25 

-    S 

+  ii 

+    10 

+  14     +  16  i  +    2 

o 

—     i 

-    4 

2 

Combined   catalogue 

1807 

-  44 

-46 

+  27 

+  36 

-38 

-  40 

-  19 

-  23    +14 

+  M 

—     2 

-    4 

3 

Bessel  II     .      .      .      . 

1823 

+     6 

+     5 

+   12 

+  IS 

+   12 

+    9 

-  32 

-  37    4-    3 

+    4 

—      I 

—      2 

4 

Struve   

1824 

+  38 

+  37 

+  16 

+  22 

+   38 

J-  oe 

-}-    4O 

-1-    -5C         AC 

—  ji 

+  10 

+         r\ 

5 

Argclander 

^     ~r 

1828 

I       J^ 

+  ii 

[     J  i 
+   10 

-  13 

-    7 

1         J^ 

+    6 

[     jj 

+    3 

+  10 

~    JO               4  J 

+    4—21 

•f-f 
—   20 

+   M 

y 

+  13 

6 

Pond  and  Airy 

1830 

+  23 

+   23 

-  35 

-  30 

-  29 

-  32 

+     3 

-    6 

—  20 

—   21 

7 

Greenwich  .... 

IS39 

-  24 

-  24 

+     I 

+     5 

-    4 

-    7 

+  18 

+  II 

+    3 

+    5 

+    3 

+      2 

S 

Greenwich  .... 

IS45 

+     i 

+      I 

+  28 

+  31 

-  18 

—  21 

+  12    +    5 

o 

+      2 

-     5 

-    5 

9 

Pulkowa     .... 

1845 

—    3 

-    3 

-  15 

—  12 

-  19 

—  22 

-  17    -  24 

+  18 

+  21 

+    S 

+    8 

10 

Greenwich  .... 

1851 

o 

o 

-  37 

-  35 

+   22 

+  IS 

+    8           o 

+  21 

+  24 

-  18 

-  18 

ii 

Greenwich  .... 

1857 

-    4 

-    3 

—    10 

-    8 

+  33 

+  29 

+  24    +15 

+  17 

+  2O 

+  10 

+    10 

12 

Paris 

1862 

o 

_j-     i 

+  18 

-t-  10 

—       J 

—     8 

4-   rfi      4-     o 

JO 

—       "7 

—      2 

2 

13 

Greenwich  .... 

1864 

+     i 

+      2 

-     5 

'    *y 

-    5 

•4 

+  18 

+  M 

i     iu      ~f™     y 

+  22      +13 

+    6 

/ 

+    9 

—   12 

-   12 

14 

Washington 

1865 

4 

-     3 

+  if) 

+  16 

+     i 

--    3 

+    f>    -  -    4 

-    5 

-      2 

+    5 

+    5 

15 

Greenwich  .... 

1869 

-    6 

-    5 

-  13 

-  13 

+    10 

+    6 

4-10            0—26 

-23 

-  18 

-  18 

16 

Washington     . 

1869 

+    7 

+     8 

-  18 

-  IS 

-     3 

-    7 

—    2    —  12  •  —  34 

i 

-  31 

+  19 

+  19 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 


Residual  corrections  given  by  catalogues. 


No. 

a  Aquiloe. 

/?  Aquilsc. 

«-  Capricorn!. 



«  Aquarii. 

a  Pegasi. 

r 

r' 

r 

r' 

r 

r 

r 

/ 

r 

r 

i 

Bradley       .... 

1755 

-    9 

—  II 

-    1 

~    9 

—  21 

-    6 

-  16 

-    5 

—      2 

—     2 

2 

Combined   catalogue 

1807 

+  15 

+  14 

-    S 

—    10 

+   10 

+  19 

+    4 

+    10 

+  2O 

+   19 

3 

Bessel  II    .... 

1823 

+    9 

+    9 

+  31 

+  29 

+  31 

+  33 

+  18 

+  23 

+    7 

+    5 

4 

Struve   

1824 

-    6 

-    6 

-     3 

+    i 

+   27 

+  34 

-    3 

4-     2 

-  29 

—  3i 

5 

Argelander 

1828 

-    3 

-    3 

-t-  II 

+    9 

-   17 

—    10 

-  17 

—  12 

+    7 

+     5 

6 

Pond  and  Airy 

1830 

»r 
/ 

-    7 

—  32 

-  34 

-  59 

-  53 

+    4 

+    8 

—      2 

-    4 

7 

Greenwich  .... 

1839 

+  23 

+  23 

—    7 

-     9 

-  27 

—  22 

-    4 

0 

+   10 

+    S 

S 

Greenwich  .... 

1845 

-    5 

-     5 

+    5 

+     3 

5 

i 

-  13 

-    10 

+      2 

0 

9 

Pulkowa    .... 

1845 

+    7 

+    7 

+     i 

—     i 

•    i 

+    4 

-    4 

I 

-    9 

—  ii 

10 

Greenwich  .... 

1851 

+     2 

+    3 

+   12 

+     9 

-  28 

-  24 

2 

+    I 

+    4 

+     i 

ir 

Greenwich  .... 

1857 

O 

+     I 

-    10 

-  13 

-  19 

-  16 

-  18 

-  16 

-     9 

-   12 

12 

1862  '  —    f. 

—    4 

—    3 

—    6 

+    S 

+  II 

—     4 

—      2 

—     i 

A 

*t 

4 

Greenwich  .... 

1864 

-     7 

-    6 

4 

-    7 

+  10 

+   12 

-  18 

-   If) 

+    4 

+        I 

14 

Washington     . 

1865 

+    4 

+    5 

-     3 

-    6 

-    5 

—     3 

+  ii 

+   13 

+  13 

+    10 

15 

Greenwich  .... 

1869 

-    S 

-    7 

+  24 

+  21 

+  10 

+   12 

-    3 

—      2 

-    4 

-    7 

16 

Washington 

1869 

-  19 

-  18 

+     7 

+    4 

•    7 

-     5 

+  25 

+  26 

+  19 

+  16 

1 

There  are  still  three  of  Maskelyne'.s  fundamental  stars  wliich  it  is  desirable,  for 
fho  sake  of  completeness,  to  add,  namely,  a  Auriga?,  a  Cvgni,  and  a.  Piscis  Anstralis. 
Although  these  stars  lie  a  little  outside  the  limits  we  have  set,  they  are  frequently  used 
as  clock-stars.  The  corrections  to  the  positions  of  the  Tabulae  Regiomontanse  were 
obtained  on  the  same  general  system  with  that  adopted  for  the  other  stars.  The; 
tabular  positions  were  corrected  for  the  terms  depending  on  the  second  and  third 
powers  of  the  time,  as  given  in  the  following  tables,  and  the  catalogue-positions  for 
systematic  error  given  by  the  formulae  of  the  last  section.  From  the  differences 
the  correction  proportional  to  the  time  was  deduced. 


RIGHT  ASCENSIONS  OF   FUNDAMENTAL   STARS. 


From  the  residuals  r  we  derive  the  folloAving  definitive  corrections  to  the  right 
ascensions  for  1 800  and  to  the  annual  variations  of  the  first  standard : 

Definitive  corrections  to  right  ascensions  of  Tabula  JKegiomontance. 


Star. 

A  R.  A. 

&y 

Definitive  correction. 

a,     Andromeda;     . 

+       5 

+       5 

-  7       +  190  T     +  o  1-  +14  T3 

y     Pcgasi  .... 

+       8 

-     26 

+  34       —164         -  6         +10 

n     Arietis  .... 

o 

—     ii 

-  3       +146         -  6         +7 

re     Ccti  

—       5 

+     17 

+  28       +98         —  6         +2 

a     Tauri     .... 

+     n 

+       8 

+  8       +98         —12         —  6 

«     Aurigx 

-   2       +26         -31         -34 

ft    Orionis. 

-       3 

+       i 

+  38         +    28           —12             -    2 

ft    Tauri     .... 

+       6 

+       9 

+  72       —156         —14         —  16 

re     Orionis. 

+        I 

+       3 

+  25              7         -  4         -  7 

re     Canis  Majoris  . 

+  155     +  29         -52          -  2       +  P 

re     Canis  Minoris. 

-15     +     8         -38         -  3       +   P 

ft    Gcminorum 

+         2 

4 

—  18       +227         +  2         —ii 

+         1 

IO 

—  17       +266         —  4         +4 

a     Lconis  .... 

i           J 
+       6 

i 

—  23       +215         —10         +  6 

ft     Leonis  .... 

7 

+       7 

—  7       +200         +   i         +10 

re     Virginis      . 

i 

-     19 

-17       +48         -  4         +8 

a     Bootis   .... 

+          2 

7 

-  5       +92         +41         +  7 

a-   Librie 

-4-           A 

+       6 

-1-8  A         —  2^S            —    6           +T 

a     Coronic 

i            H 
+           & 

i? 

I     u-f                    ^"33                                                 '        J 

-  4       +135          -  6         +3 

a     Scrpcntis    . 

+         2 

-       5 

+  50       —141         —  10         +   i 

re     Scorpii  .... 

IO 

+     15 

+  31       —  116         —10          -  9 

re     Ilerculis 

+         2 

+       3 

-3       +69          -8          -2 

n     Ophiuchi     . 

+       3 

+     10 

—  24       +360         -  3          -  2 

re     Lyric      .... 

0 

4 

+    2         +S2           —26             -    I 

}-     Aquilac  .... 

+          2 

3 

+  71       —197         —10          -  i 

«,     Aquilx  .... 

+          I 

-       3 

+  8       +32         —20          -   i 

ft     Aquilrc  .... 

+         2 

+        i 

+41       —215         +5          -   i 

re'-   Capricorn!  . 

10 

+       12 

—  16       +     6         —12         —  3 

re     Cygni    .... 

-32       +184         -17         +  3 

n     Aquarii. 

7 

+       8 

-41       +173          -  7         +  6 

n     Piscis  Australis    . 

—  50       —152         -  6         +11 

re     Pcgasi  .... 

+       i 

+       3 

+  10       +21          -  4         +9 

In  the  above  table  P  signifies  the  periodic  part  of  Auwers'  correction,  which  is 
more  convenient  to  use  than  q. 

The  following  table  gives  the  definitive  right  ascensions  for  every  five  years  from 
1750  to  1900. 

I  conceive  that  the  mean  error  of  these  right  ascensions,  after  correction  for 
equinox,  will  not  exceed  os.oio  at  any  time  during  the  nineteenth  century. 


DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS.  53 

Mean  right  ascensions  for  beginning  of  each  fiflli  Bcssdlan  fictitious  year,  \  750-1850. 


Year. 

a  ANUKOMEDJE. 

y  PKGASI. 

K.  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

//.  m.    s. 

s. 

/i.  in.   s. 

s. 

'/So 

23  55  31-079 

3  •  0665 

o  o  23.520 

3.0702 

i/55 

46.413 

3.0674 

38.872 

3.0707 

1760 

23  56  1.752 

3.0683 

54-  227 

3.0712 

1/65 

17-095 

3-0691 

o  i  9.584 

3.0716 

1770 

32.443 

3.0700 

24-943 

3.0721 

'775 

47-795 

3.0709 

40.  305 

3.0726 

1780 

23  57  3-  I51 

3-0717 

55-669 

3-0730 

1785 

18.512 

3.0726 

0   2  11.035 

3-0735 

1790 

33-876 

3-0735 

26.  404 

3.0740 

'795 

49.  246 

3  •  0744 

41-775 

3-0745 

1800 

23  58  4.620 

3-0/53 

57-  148 

3-0749 

1805 

19.998 

3-0761 

o  3  12.524 

3-0754 

1810 

35-38i 

3-0770 

27.902 

3-0/59 

1815 

50.  768 

3-0779 

43-  283 

3.0764 

1820 

23  59  6.  i  60 

3.0788 

58.666 

3.0769 

1825 

21.  556 

3-0797 

o  4  14.052 

3-0774 

1830 

36.  956 

3  .  0806 

29.440 

3-0778 

•«35 

52.361 

3-0815 

44-830 

3-0783 

1840 

o  o  7.771 

3.0824 

050.  223 

3.0788 

1845 

23-  185 

3-0833 

15.618 

3-0793 

1850 

38-603 

3.0842 

31.016 

3.0798 

1855 

54.026 

3-0851 

46.  416 

3  •  0803 

1860 

o  i  9.454 

3  .  0860 

o  6  1.819 

3.  0808 

1865 

24.886 

3  .  0869 

17.224 

3-0813 

1870 

40.323 

3.0879 

32.632 

3.0818 

1875 

55-764 

3.0888 

48.042 

3-0823 

1880 

O   2  I  I.  2IO 

3-0897 

-o  7  3-455 

3.0828 

1885 

26.  661 

3.0906 

18.870 

3-0833 

1890 

42.  1  16 

3-0915 

34.288 

3.0838 

1895 

57-575- 

3-0925 

49.  708 

3-0843 

1900 

o  3  13.040 

3-0934 

o  8  5.131 

3.0848 

54 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

Mean  riyM  ascensions,  d'c. — Continued. 


Year. 

a  ARIKTLS. 

a.  CETI. 

R.  A. 

Ann.  Viir. 

R,  A. 

Ann.  Vav. 

//.    'ill.            S. 

S. 

h.  m.        s. 

s. 

1750 

i   53     8.365 

3-3430 

2  49   14.423 

3-  1170 

1755 

25-083 

3-3440 

30.  009 

3-  H75 

1760 

41.805 

3-3450 

45-598 

3-  H79 

1765 

58.533 

3-346o 

2     50        I.  189                     3.  1184 

1770 

i   54   15.266 

3-3470 

16.  782              3.  1  189 

1775 

32.003 

3-348o 

32.377              3-II94 

1780 

48.  746 

3-3490 

47-975              3-1198 

1785              i   55     5-493 

3-3500 

2   51     3-576              3.  1203 

1790 

22.245 

3-3510 

19.  179 

3.  1208 

1795 

39.003 

3-3520 

34-  784 

3-  1213 

1800 

55-765 

3-3530 

50-391 

3.1218 

1805              i   56   12.533 

3-3540 

2   52     6.001 

3  .  1222 

1810                        29.305 
1815                        46.082 

3-3550 
3-356o 

21.614 

37.228 

3-  !227 

3-  1232 

1820              i   57     2.865 
1825                        i9-652 
1830                        36.445 

3-3570 
3-358o 
3-3590 

52.846 

2  53     8.465 
24.087 

3-  1237 
3-  I242 
3.  1246 

1835                        53-242 

3.3600 

39.712 

3-  1251 

1840              i   58   10.045 

3.3610 

55-338 

3-  1256 

1845 

26.852 

3.3620 

2   54   10.968 

3.  1261 

1850 

43-665 

3-3630 

26.  599 

3.  1266 

1855 

i   59     0.482 

3-3640 

42.233 

3-  1271 

1860 

17-305 

3-3650 

57-870 

3-1275 

1865 

34-  133 

3.3661              2  55   13.509 

3.  1280 

1870 

50.966 

3-3671 

29-  150 

3-1285 

1875 

2     o     7.803 

3-368! 

44-  794 

3-  I290 

1880 

24.  646 

3-3691 

2  56     0.440 

3-  1295 

1885 

41.494 

3-37oi 

16.088 

3-  1300 

1890 

58.347 

3-3711 

31-739 

3-  1304 

1895 

2        115.  2O6 

3-3721 

47-393 

3-  1309 

1900 

32.  069 

3-3732 

2  57     3-049 

3-  i3H 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  right  ascensions,  &c. — Continued. 


55 


Year. 

a  TAUKI. 

a  AURIGA. 

11.  A. 

Ann.  Var. 

11,  A. 

Ann.  Var. 

//.  III.      «S. 

s. 

li.  in.   ,s. 

s. 

i/50 

4  21  36.293 

3-4227 

4  58  15-977 

4.4004 

i/55 

53-408 

3-4232 

37-98i 

4.4014 

i  760 

4  22  IO.  526 

3-4238 

59.990 

4.4023 

1/65 

27.  646 

3-4243      4  59  22.004 

4.4032 

1770 

44-  769 

3-4248          44-023 

4.4042 

'775      4  23  1.894 

3-4254 

5  o  6.046 

4.4051 

i  780          19.  022 

3-4259 

28.073 

4  .  4060 

1785          36-  'S3 

3-4265 

50.  106 

4.4069 

1790          53  .286 

3.4270      5   i  12.  142 

4.4078 

1795      4  24  10.423 

3-4275 

34-184 

4.4087 

1800          27.562 

3.4281 

56-230 

4.4096 

1805          44-704 

3-4286 

5  218.  280 

4.4105 

1  8  1  o      4  2  5   i  .  848 

3-4291 

40.335 

4.4114 

1815          18.995 

3.4296 

5  3  2.394 

4.4122 

1820          36.144 

3-4302 

24-457 

4-4I31 

'825          53-297 

3-4307 

46.525 

4.4140 

1830      4  26  10.452 

3-4312 

5  4  8.597 

4.4148 

1835          27.609 

3.43i8 

30.673 

4-4'57 

1840          44.  769 

3.4323 

52.754 

4.4165 

1845      4  27   1.932 

3-4328 

5  5  14-839 

4-4174 

1850 

19.  097 

3-4333 

36.928 

4.4182 

1855 

36.  265 

3-4339 

59-022 

4.4190 

1860 

53-436 

3-4344 

5  621.119 

4.4199 

1865 

4  28  To.  609 

3-4349 

43-  221 

4.4207 

1870 
•8/5 

'27-  785 
44.963 

3-4354 
3-4359 

5  7  5-326 
27.436 

4-4215 
4.4223 

1880 

4  29  2.  144 

3-4364 

49-550 

4-4232 

1885 

'9-327 

3-4370 

5  8  11.668 

4.4240 

1890 

36.5'4 

3-4375 

33-790 

4-4248 

'<S95 

53-702 

3.438o 

55-9I6 

4.4256 

1900 

4  30  10.894 

3-4385 

5  9  18.045 

4.4264 

RIGHT   ASCENSIONS  OF   FUNDAMENTAL   STARS, 

Mean  riylit  ascensions,  &c. — Continued. 


/?  ORIONIS. 

/?  TAUEI. 

1  p'll* 

-1  Cd  1  • 

R.  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

li.  'in.   s. 

s. 

h.  m.    s. 

8. 

1750 

5  2  32.084 

2-8757 

5  10  30.599 

3-7775 

1755 

46.463 

2-8759 

49.488 

3-  7779 

1760 

5  3  o.  843 

2.8761 

5  ii  8.378 

3-  7784 

1765 

15.224 

2.8763 

27.  272 

3-7788 

1770 

29.  606 

2.8765 

46.  167 

3-7793 

•775 

43.989 

2.8767 

5  12  5-064 

3-  7797 

1780 

58.373 

2.  8769 

23.964 

3-  7802 

1785      5  4  12.758 

2.8771 

42.  866      3.  7806 

1790 

27.  144 

2.8773 

5  U   1-770 

3.7811 

T  795 

4i.53i 

2.8775 

20.  677 

3-7815 

1800 

55.9I9 

2.8777 

39.585 

3./8.9 

1805 

5  5  10-309 

2.8779 

58.496 

3-  7824 

1810 

24.699 

2.8781 

5  14  17.409      3.  7828 

1815 

39.090 

2.8783 

36.324 

3-7832 

1820 

53.482 

2.8785 

55-24I 

3.7836 

1825 

5  6  7.875      2.8787 

5  i5  H.  160 

3.7840 

1830 

22.  269         2.  8789 

33-o8i 

3-  7845 

1835 

36.665         2.8791               52.005 

3-7849 

1840 

51.061         2.8793 

5  16  10.930 

3.7853 

1845 

5  7  5-458 

2.8795 

29-858 

3-7857 

1850 

19.856 

2.8797 

48.787 

3.7861 

i855 

34-256 

2-8799      5  17  7.719 

3-7865 

1860 

48.656 

2.8801 

26.  652 

3-7869 

1865 

5  8  3.057 

2.8803 

45-588 

3.7873 

1870 

17-459 

2.8805 

5  i8  4-526 

3.7877 

1875 

31.862 

2.8807 

23-465 

3.7881 

1  880 

46.  267 

2.8809 

42.407 

3-7885 

1885  ' 

5  9  0.672 

2.8811      5  19  1.350 

3.7889 

1890 

15.078 

2.  8813               2O.  296 

3-7893 

1895 

29.485 

2.8815 

39-243 

3-7897 

1900 

«.893 

2.8817 

58.  193 

3-  790, 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  riyJtt  ascensions,  cCr. — Continued. 


57 


a  OKIOXIS.              a  CANIS  MAJOKIS.* 
Year 

\{.  A. 

f 

Ann.  Yiir.        U.  A. 

Ann.  Var. 

//.  m.   s. 

s. 

//.  m.   s. 

s. 

1750      5  41  38.680 

3.2429 

6  34  7.882 

2.  6446 

i/55          54-895 

3-2431 

21-  105 

2.6445 

i  760      5  42  1  1  .  1  1  1 

3-2432 

34-327 

2.6445 

1765          27.327 

3-2434 

47-550 

2-6445 

!/7o          43-545 

3-2435      6  35  0.772 

2-6444 

'775          59-763 

3-2437          13-994 

2-6444 

1780      5  43  15.982 

3-2439 

27.  216 

2.6444 

1785          32.201 

3.2440 

40.438 

2  -  6444 

1790          48.422 

3.2442          53-66o 

.  2.6443 

1795      5  44  4-643 

3-2443 

6  36  6.  88  i 

2-6443 

1800          20.865 

3-2445 

2O.  IO2 

2-6443 

1805          37-088 

3-  2446 

•">  1    1  *>  1 

oo-  j-4 

2.6442 

1810          53-311 

3.2448 

46.545 

2.  6442 

•815.     5  45  9-535 

3-  2449 

59-766 

2.6442 

1820          25.  760 

3-  245' 

6  37  12.986 

2.6441 

1825          41.986 

3-2452 

26.  207 

2.6441 

1830          58.212 

3-2454 

39-427 

2.6441 

1835      5  46  H-439 

3-2455 

52.647 

2.  6440 

1840         30.667 

3-2456 

6  38  5-867 

2  .  6440 

1845         46.896 

3-2458 

19.087 

2.6440 

1850      5  47  3.  125 

o-  2459 

32.307 

2-6439 

1855          '9-355 

3.2461 

45-527 

2-6439 

1860          35-586 

3.2462 

58.  746 

2.6439 

1865          51-817 

3.2464 

6  39  11-965 

2-6438 

1870      5  48  8.050 

3-2465 

25.  184 

2-6438 

1875          24.282 

3.2466 

38-403 

2.6437 

1880          40.516 

3.2468 

51.622 

2-6437 

1885          56-750 

3.2469 

6  40  4.  840 

2.6437 

1890      5  49  12.985 

3.2470 

18.058 

2-6436 

1895         •  29.  220 

*>   It**'* 

j-  -4/  - 

31.276 

2.6436 

1900          45-457 

•^   "J  \  *7  ^ 

o-  -4/0 

44.494 

2-6436 

8— F  S 


The  corrections,  P,  on  p.  Gl>,  are  to  l>o  applied  to  tliese  positions  of  a  Canis  Mnjoris. 


RIGHT   ASCENSIONS  OF   FUNDAMENTAL   STARS. 

Mean  right  ascensions,  £c. — Continued. 


Year. 

a  CANIS  MINORIS.* 

/?  G-EMINORUM. 

R.  A. 

Ann.  Var. 

II.  A. 

Ann.  Var. 

/;.  m.        s. 

s. 

n.  m.     s. 

.    s. 

1750 

7   26   12.027 

3-I507 

7  29  58.793 

3.6964 

J755 

27.780 

3-  1505 

7  30  17-  274 

3-6958 

1760 

43-532 

3-  J502 

35-751 

3-6952 

1765 

59.282 

3-  H99 

54-225 

3.6946 

1770 

7   27   15.031 

3-  H97 

7  31    12.697 

3.6940 

!775 

30.779 

3-  H94 

31-165 

3-6934 

1780 

46.525 

3-  H92 

49.631 

3-6928 

1785 

7  28     2.  271 

3-  1489 

•  7  32     8.093 

3.6922 

1790 

18.015 

3-  1487 

26.553 

3.6916 

1795 

33-757 

3-  1484 

45-009 

3.6910 

1800 

49-499 

3.1482 

7  33     3-462 

3  •  6904 

1805 

7  29     5.239 

3-  H79 

2*1.913 

3.6898 

1810 

20.978 

3-  J476 

40.  360 

3.6892 

1815 

36-715 

3-  1474 

58.804 

3.6885 

1820 

52-452 

3-  i47i 

7  34   17-246 

3-6879 

1825 

7  30     8.187 

3-  H69 

35-684 

3-6873 

1830 

23.920 

3.  1466 

54-  H9 

3.6867 

1835 

39-653 

3-  1463 

7  35    12.551 

3.6861 

1840 

55-384 

3-  H6i 

30.980 

3-6855 

1845 

7  3i    ii.  H3 

3-  HSS 

49.405 

3.6848 

1850 

26.842 

3-  H56 

7  36     7-828 

3.6842 

1855 

42.569 

3-  H53 

26.248 

3.6836 

1860 

58.295 

3-  H5o 

44.  664 

3-6829 

1865 

7  32   14.019 

3-  1448 

7  37     3-077 

3-6823 

1870 

29.742 

3-  1445 

21.487 

3.6817 

1875 

45.464 

3-  H42 

39-894 

3.6810 

1880 

7  33     1-185 

3.1440 

58.298 

3  •  6804 

1885 

1  6  .  904 

3-  H37 

7  38  16.699 

3-6798 

1890 

32.  622 

3-  H34 

35-097 

3.6791 

1895 

48.338 

3-  M32 

53-491 

3-6785 

1900 

7  34     4-053 

3-  H29 

7  39   11-882 

3.6778 

*  The  corrections,  P,  on  p.  71,  arc  to  bo  applied  to  these  positions  of  a  Canis  Minoris. 


DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  riylit  ascensions,  (fr.— Continued. 


59 


Year. 

a  HYDRJE. 

a  LEONIS. 

R  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

I/.  III.     S. 

8. 

//.  m.    s. 

s. 

1750      9  15  17.906 

2.9512 

9  55   1-762 

3-  2148 

'755 

32.662 

2.9512 

17-835 

3-2143 

1760          47.418 

2-95'i          33-905 

3-2138 

1765      9  16  2.  173 

2.9510          49-973 

3-2133 

1770          16.928 

2-9509      9  56  6.038 

3.2128 

1775 

31.682 

2.9508 

22.  IOO 

3  .  2122 

1780          46.436 

2-9507 

38.  I  60 

3.2II7 

1785      917   i.  190 

2-9507 

54-217 

3.  21  12 

'79°          '5-943 

2.9506 

9  57  10.272 

3.2107 

'795 

30.  695 

2-9505 

26.324 

3.  2102 

1800 

45.448 

2.9504 

42-373 

3.  2096 

1805 

918   O.  2OO 

2-9503 

58.420 

3  .  209  i 

1810 

'4-951 

2-9503 

9  S8  14-465 

3.  2086 

.815 

29.  702 

2-  95O2 

30-507 

3.  2081 

1820 

44-453 

2.9501 

46.  546 

3.2076 

1825 

59-203 

2-9500 

9  59  2.582 

3-  2071 

1830 

9  19  '3-953 

2.9500 

18.617 

3.  2066 

'835 

28.  703 

2-9499 

34.648 

3.  2061 

1840 

43-452 

2.9498 

50.677 

3.2056 

1845 

58.  2or 

2-9497 

10  o  6.  704 

3.2050 

1850 

9  20  12.950 

2-9497 

22.  728 

3-2045 

•855 

27.698 

2.9496 

38.749 

3.2040 

1860 

42.446  ' 

2-9495 

54-  768 

3-2035 

1865 

57-  '93 

2-9495 

10  i  10.  784 

3.2030 

1870 

921  ir.  940 

2  .  9494 

26.798 

3.2025 

i875 

26.687 

2-9493 

42.  Sio 

3.  2020 

1880 

41-433 

2.9492 

58.818 

3-  2015 

1885 

56-  179 

2.9492 

IO   2  14.825 

3-  2OIO 

1890        9  22  IO.  925 

2.9491 

30.829        3-2005 

'895 

25.670 

2  .  9490 

46.  830        3.  2OOO 

I9OO 

40.416 

2-9490' 

10  3  2.829 

3-  '995 

6o 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 

Mean  riff  Jit  ascensions,  dec. — Continued, 


Year. 

/?  LEONLS. 

a  VIEGINIS. 

R  A. 

Ann.  Var. 

R  A. 

Ann.  Var. 

Ii.   m.        s. 

s. 

/(.    In.         s. 

s. 

1750            ii    36   17.250 

3.0745                  13     12        3.448 

3-i38i 

1755                         32.622              3-0741 

19.  140 

3-*3S6 

1760                        47-992              3-0737 

34-834 

3-  1392 

1765             **    37     3-36o              3-0733 

50.531 

3-  *397 

1770                         18.7-25              3-0729 

13    13     6.231 

3-  H03 

1775 

34.089              3-0725 

21-934 

3-  1408 

1780 

49-45*              3.0722 

37-639 

3-  i4H 

1785 

ii   38     4.811              3-0718 

53-348 

3-  Hi9 

1790 

20.  169              3.  0714 

13    *4     9-059 

3-  H25 

1795                        35-525              3-07*o 

24.772 

3-i430 

1800 

50,879 

3.0706 

40.  489 

3-  *436 

1805 

ii    39     6.231 

3.  O7O2                                    56.  2O8 

3-144* 

1810 

21.581               3-0699            13    15    11.930 

3-*447 

1815 

36.929              3-0695                         27.655 

3-  1452 

1820 

52.276              3-0691 

43-382 

3-  H58 

1825            ii   40     7.620             3.0687 

59-ii3 

3-  *464 

'1830                         22.963              3-0683       |      13    16   14.846 

3-  1469 

1835 

38.304              3.0680 

30.582 

3-H75. 

1840 

53-642 

3.0676 

46.321 

3-  1481 

1845             ii   41      8.980 

3.0672 

13    17     2.063 

3:*486 

| 

1850 

24-315    ' 

3.0669 

17.807 

3-  1492 

1855 

39.648 

3.^0665 

33-  555 

3-  *498 

1860 
1865 

54.980 
1  1   42    10.  309 

3-  O66l 
3.0658 

49-305              3-1503 
13   i  8     5.058              3-1509 

1870                         25.637 

3-0654 

20.814             3-15*5 

1875                         40-963 

3.0650 

36.573              3-1520 

1880                         56.288 

3.0647 

52-334             3-1526 

1885             ii   43    11.610 

3-0643 

13    19     8.099             3-  iSS2 

1890                         26.  931 

3.0640 

23.866             3.1538 

1895                         42.250 

3.0636 

39-637              3-1544 

1900                         57-567 

1 

3-0633 

55-4io 

3-*549 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  right  ascensions,  dr. — Continued. 


61 


Year. 

a  Boons. 

or  LIBRAE. 

K.  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

//.    111.     .V. 

s. 

//.   ill.    S. 

s. 

i/5o 

14  4  15.978      2.7320 

'4  37  5-726 

3.2881 

'755 

29-638      2.7321 

22.  I  68 

3.2888 

1760 

43-299      2.7322 

38.614 

3.2896 

1765 

56.960      2.7323 

55.064 

3.2904 

1770 

14  5  10.622      2.7324 

14  38  11.518    3.2911 

1/75 

24.284      2.7325 

27.976 

3-  2919 

1780 

37.946      2.7326 

44-437 

3-2927   . 

i  785 

51.609      2.7327 

14  39  0.902 

3-2934 

1790  . 

14  6  5.273      2.7328 

17-37I 

3.2*942 

'795 

* 

l8-937      2.7329 

33.844 

3-  2950 

1  800 

32.601      2.7330 

50-321 

3-2957 

.805 

46.266      2.7331 

14  40  6.  802 

3-2965 

1810 

59-932      2.7332 

23.286 

3-2973 

1815 

14  7  13.598      2.  7333 

39-  775 

3-  2980 

1820 

27.265      2.7334 

56.267 

3.2988 

1825 

40.932      2.7335 

14  41  12.  763 

3.2996 

1830 

54-599      2.7336 

29.263 

3-3004 

1835 

1840 

14  8  8.267      2.7337 
21.936      2.7338 

45-  766 
14  42  2.274 

3-30ii 
3-3QJ9 

1^45 

35-6o5 

2-7339 

18.785 

3.3026 

.850 

49-275 

2.  7340 

35-301 

3-3034 

1855 

14  9  2.945 

2-7341 

51.820 

3-3042 

1860 

16.616 

2-7342 

14  43  8.343 

3-3050 

,865 

30.  288 

2-  7344 

24.870 

3-3058 

1870 

43.960      2.7345 

41  .  401 

3-3065 

i875 

57-633      2.7346 

57-935 

3-3073 

1880 

14  i  o  11.  306 

2-7347 

14  44  14.474 

3-3081 

,885 

24.  980 

2-7348 

31.016 

3-3089 

1890 

38-654 

2-7350 

47-562 

3.3096 

'895 

52.329 

2-7351 

14  45  4-  U3 

3-3104 

1900 

14  ii   6  .  005 

2-7352 

20.  667 

3-3"2 

62 


RIGHT   ASCENSIONS   OF    FUNDAMENTAL   STARS. 

Mean  right  ascensions,  rfr. — Continued. 


a  CORONA. 
Yo-u- 

a  SERPENTIS. 

J_  CCliJ.  . 

K  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

/;.  in.   s. 

s. 

//.  111.     S. 

s. 

1750     15  24  6.587 

2.5360 

J5  3i  58.355 

2.9431 

i/55          19.267 

2-5361 

15  32  13-071 

2-9434 

1760          3I-948 

2.5363 

27.789 

2-9437 

1765          44-630 

2-5364 

42.508 

2  .  9440 

i//o          57-312 

2-5365          57-229 

2-9443 

1775     J5  25  9-994 

2-5366     15  33  11.952 

2.9446 

1780          22.678 

2.5367          26.675 

2.9449 

'785          35-362 

2.5368          41.401 

2.9452 

1790          48.046 

2.5369          56.128 

.2-9455 

1795     15  26  0.731 

2-5370     '5  34  10.856 

2.9458 

• 

i  800          13-416 

2-5372 

25-586 

2.  9462 

1805               26.  IO2 

2-5373 

40.318 

2.9465 

1810          38.789 

2-5374          55-05I 

2.9468 

1815          5^476 

2-5375     15  35  9-  785 

2.9471 

1820     15  27  4.  164 

2.5376          24.521 

2-9474 

1825          16.852 

2-5377 

39-259 

2-9477 

1830          29.541 

2.5378 

53.998 

2.  9480 

1835          42.231 

2-5380     15  36  8.739 

2.9483 

1840          54-921 

2.5381          23.481 

2.  9486 

1845     '5  28  7-612 

2-5382 

38-225 

2.9489 

1850          20.303 

2-5383 

52.970 

2.9492 

1855          32.995 

2-5384     15  37  7-7I7 

2-9495 

1860          45.  687 

2.5386          22.465 

2.9498 

.865          58.380 

2-5387          37-215 

2.9501 

1870     15  29  11.074 

2-5388          5^966 

2.9504 

1875          23.768 

2-5389     i5  38  6.719 

2.9507 

1880          36.463 

2.5390          21.473      2.9510 

1885          49-159 

2.5392 

36.  229 

2-95'3 

1890     15  30  1.855 

2-5393          50-987 

2.9516 

1895          H.552 

2-5394     15  39  5-746      2.9520 

1900          27.  249 

!          'I 

2-5395 

20.506      2.9523 

i 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  rigid  ascensions,  (C-c. — Continued. 


Year. 

a  ScORPIl. 

a  HERCUIJS. 

R.  A. 

Ann.  Var. 

R.  A. 

Ann.  Var. 

//.  in.   t>. 

s. 

h.  in.   s. 

6'. 

1750 

1  6  14  7.423 

3.6487     17  3  15.554 

2.7286 

!755 
1760 

25.669 

43.918 

3-6495 
3-6503 

29.  198 
42.842 

2.  7288 
2.  7290 

r/65 

1615   2.  172 

3-6511 

56.487   • 

2.7291 

1770 

2O.  429 

3-6519 

'7  4  10.  133 

2.7293 

1775 

38.690 

3-6527 

23.780 

2-  7295 

1780 

56.956 

3-6534 

37-428 

2.  7297 

1785 

16  16  15.  225 

3-6542 

51-077 

2.7299 

1790 

33-498 

3-6550 

17  5  4-727 

2.7300 

i/95 

51-775 

3-6558 

18.378 

2.7302 

1800 

16  i  7  10.  056 

3.6566 

32.029 

2.7304 

1805 

28.341 

3.6573 

45.681 

2.7306 

1810 

46.  629 

3-6581          59-335 

2.7307 

1815 

16  18  4.922 

3-6589 

17  6  12.989 

2.  7309 

1820 

23.218 

3-6596 

26.  644 

2.7311 

1825 

41.518 

3.6604 

40.  300 

2-  73'3 

1830 
1835 

59-  822 
16  19  18.  130 

3.  66  i  2 
3.6619 

53.956 
17  7  7.614 

2.7314 
2.7316 

1840 

36.442      3-6627 

21.  273 

2.7318 

1845 

54-757 

3-6635 

34-932 

2.7320 

1850 

16  20  13.076 

3.6642 

48.592 

2.7321 

'855 

3'  -399 

3-6650 

17  8  2.253 

2.7323 

1860 

49-  726 

3-6657 

I5-9I5 

2.7325 

1865 

16  21  8.057      3-6665 

29-578 

2.7327 

1870 

26.391 

3-6673 

43.242 

2.7328 

'875 

44-  730 

3.6680 

56.906 

2.7330 

1880 

l6  22   3.O72 

3.6688 

17  9  10.572 

2.7332 

1885 

21.417 

3-6695 

24.238 

2-7333 

1890 

39.767 

3-6703 

37-905 

2-7335 

'895 

58.  120 

3.6710 

51-573 

2-7337 

1900 

16  23  16.477 

3.6718 

17  10  5.  242 

2-7339 

RIGHT  ASCENSIONS  OF   FUNDAMENTAL  STARS. 

Mean  rujld  ascensions,  <('-c. — Continued. 


Year. 

a  OPHIUCIII. 

a  LYRJE. 

11.  A. 

Ann.  Var.       K.  A. 

Ann.  Var. 

li.  m.   s. 

s.       li.  m.    s. 

,5. 

1750 

17  23  20.430 

2.7781     18  28  28.585 

2.  0298 

1755 

34-320 

2.7783 

38.734 

2.  0299 

i  760 

48.  212 

2-7784 

48.883 

2.  0299 

1765 

17  24   2.  105 

2.7786 

59-033. 

2  .  O3OO 

1770 

15.998 

2.7788 

18  29  9.  183 

2  .  O3OO 

1775 

.   29.893 

2.7790 

19-334 

2.0301 

1780 

43-788 

2.  7791 

29.484 

2.  O3OI 

1785 

57.684 

2-7793 

39-635 

2.0302 

1790 

17  25  II.58I 

2-7795 

49.786 

2.0303 

1795 

25-479 

2-7797 

59-938 

2.0303 

1800 

39-377 

2.7798 

18  30  10.089 

2  .  0304 

1805 

53-277 

2.  7800 

20.  241 

2.0304 

1810 

17  26  7.  177 

2.7802 

30-  394 

•2.0305 

1815 

21.079 

2.7803 

40.  546 

2.0305 

1820 

34.981 

2.7805 

50.699 

2.  0306 

1825 

48.884 

2.7807 

18  31  0.852 

2.  0306 

1830 

17  27  2.788 

2.  7809 

i  i  .  005 

2.0307 

1835          16.692 

2.  7810 

21.  159 

2.0307 

1840          30-598 

2.  7812 

3I-3I3 

2.  0308 

1845 

44-  504 

2.7814 

41.467 

2.  0308 

1850 

58.411 

2.7815 

51  .  621 

2.0309 

1855 

17  28  12.319 

2.7817 

18  32   1.776 

2.  O3IO 

1860 

26.  228 

2.7819 

11.931 

2.  O3IO 

1865- 

40.  138 

2.  7820 

22.  086 

2.03II 

1870 

54.049 

2.  7822 

32.241 

2.03II 

1875 

17  29  7.960 

2.7824 

42-397 

2  .  03  I  2 

1880 

21.872 

2.7825 

52.553 

2.0312 

1885 

35-785 

2.  7827 

i  8  33  2.709 

2.0313 

1890 

49.699 

2.  7829 

12.865 

2.0313 

1895 

17  30  3.614 

2./830 

23.  022 

2.0314 

1900 

17-530 

2.7832 

33-  179 

2.03H 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  riyld  ascensions,  dr. — Continued. 


Yea  r. 

y  AQI:IL 

JE. 

a  AQUIL 

£.. 

K.  A. 

Ann.  Yar. 

R.  A. 

Ann.  Yar. 

//.  HI.     S. 

s. 

//.   1)1.     S. 

S. 

'750 

19  54  22.396 

2-8536 

'9  38  34-939 

2.9502 

1755 

36.664 

2-8536 

49-590 

2.9501 

i  760 

50.932 

2-8535 

19  39  4.  240 

2.9500 

1/65 

19  35  5-  200 

2-8535 

18.890 

2.9299 

'770 

19.467 

2-8554 

33-539 

2.  9298 

'775 

->  t  **  ->  \ 
jj>-  734 

2-8534 

48.  i  88 

2.9297 

1780 

48.  ooi 

2-8534 

19  4O   2.  856* 

2.  9296 

-785 

19  36    2.  268 

2-8533 

17  484 

2.9295 

1790 

16.534 

2-8533 

32-  132 

2.9295 

'795 

30.  800 

2-8532 

46.  779 

2.9294 

iSoo 

45.066 

2-8532 

19  41   1.425 

2.9295 

1805 

59-332 

2-8551 

16.  071 

2.9292 

i  8  10 

'9  37  '3-597 

2-853' 

30.  7'  7 

2.9291 

1815 

27.865 

2-8530 

45-362 

2.  9290 

1820 

42.  128 

2-8530 

1942  o.  007 

2.  9289 

1825 

56-392 

2-8529 

14.652 

2.9288 

1830 
1835 

19  38  10.657 
24.921 

2-8529 
2.8528 

29.  296 
•43-939 

2.9287 
2.9286 

1840 

39-  185 

2.8528 

58-582 

2.9286 

i*45 

53-448 

2-8527 

'9  43  13-225 

2.9285 

1850 

19  39  7.  712 

2.8527 

27.867 

2.9284 

t855 

21-975 

2.8526 

42.508 

2.9285 

1860 

36.  238 

2.8526 

57-  150 

2.  9282 

1865 

50.  500 

2-8525 

i  9  44  i  i  .  790 

2.  9281 

1870 

19  40  4.763 

2.8524 

26.450 

2.  9280 

1^75 

19.025 

2.8524 

41.  070 

2.9279 

1  880 

33-287 

2-8523 

55-7'Q 

2.9278 

1885 

47-548 

2-8523 

19  45  10.349 

2.9277 

1890 

1941   i  .  810 

2.8522 

24.987 

2.  9276 

1895 

i  6.  071 

2.8522" 

39-625 

2.9275 

1900 

**  C\   ">  *>  T 

0U-  JJ1 

2.8521 

54.262 

2.9274 

9 — F  s 


66 


RIGHT  ASCENSIONS   OF   FUNDAMENTAL   STARS. 

right  ascensions,  <fr. — Continued. 


/?  AQUILJE. 

a2  CAPRICORNI. 

1  car. 
K.  A. 

Ann.  Var. 

H.  A. 

Ann.  Var. 

//.  111.     S. 

s. 

Ji.  m.    s. 

s. 

1750     19  43  1-8/4 

2.  9490 

20  4  9-754 

3-3439 

1755          16.619 

2.9490 

26.473 

^>   ^  1  ^  C 

o  •  J4o  0 

1760          3i-364 

2.9489 

43  •  i  89 

^   ^  1  ">  T 

o  •  34.5  l 

i  765          46.  108 
1770     19  44  0.852 

2.9488 
2.9488 

59.902 
20  5  16.614 

3-3426 
3-3422 

'775          15-595 
1780          30.339 

2.9487 
2.9486 

33-324 
50.032 

3-34I8 
3-34I4 

1785          45-082 

2.  9486     20  6  6.  738 

3-3410 

1790          59-  824 

2.9485 

23-442 

3  •  34o6 

'795     T9  45  U-567 

2  .  9484 

40'.  144 

3-3402 

1800          29.308 

2.9483 

56-844 

"*  ">  **  c\  — 

o-  o  o9/ 

1805          44.050 

2.9483     20  7  13.542 

">  ^  ^  /™i  t 
o  •  oo9o 

1810          58.  791 

2.9482 

30-237 

3-3389 

1815     19  46  I3-532 

2  .  948  I 

46.931 

3-3385 

1820          28.  273 

2.9481 

20  8  3.622 

1   1  1<s  T 

o-  33°  ' 

'825          43-oi3 

2.  9480 

20.311 

3-3376 

1830          57-752 

2-9479 

36.999 

1   ••»  ->  *7  *> 

o-  o  j/  - 

'835      19  47  12.492 

2-9479 

53-684 

->  1  1^9 
3-  oo°b 

1840           27.231 

2.9478 

20  910.  367 

0    -^  O  (•)  1 

o-  oo°4 

1845           41.970 

2-9477 

27.048 

•*>  *>  ^  /Sr^ 
0-  JO00 

1850           56.  708 

2.9476 

43-726 

3  •  3355 

1855      19  48  11.446 
1860          26'.  184 

2.9476 
2-9475 

20  10  0.403 
17.078 

*>"   ^>  'I  f  T 

O-  JO5  * 

-1   o  ^>  /I  *• 

o-  jo4/ 

1865          40.  921 

2.9474 

33-750 

^>   -l  ^  -1  1 

o  •  jo4o 

1870          55-658 

2-9473 

50.421 

o   ^  o  o  r\ 

j  •  ooo9 

1875     19  49  10.394 

2-9473 

20  i  i   7.  090 

3-  3334 

1880          25.131 

2.9472          23.756 

3-3330 

1885          39.866' 

2.9471 

40.  420 

3-3326 

1890          54.602 

2.9471 

57.082 

i   o  t  o  O 

o  •  o  J  -  - 

l895     19  50  9-337 

2.9470 

2O  12  13.  742 

I 

o   "»  ^y  T  *7 
D-  O  J  j  / 

1900          24.  072 

2  .  9469 

30.  400 

•*>   o  •I  T  ^ 

o-  oo  j  o 

DEFINITIVE  RIGHT  ASCENSIONS  OF  THE  FUNDAMENTAL  STARS. 

Mean  riyld  ((scomions,  iCc. — Continued. 


67 


Year. 

a  CYGXI. 

a  AQKAKII. 

R.  A. 

Ann.  Yar. 

R.  A. 

Ann.  Yar. 

//.   III.     S. 

S. 

//.  III.     S. 

s. 

I750       20  32  54-957 

2.0414 

21  52  56.049 

3.0888 

1  755 

-o  33  5-  H4      2.0415 

21  55  11.492      5.0886 

i  760 

15.352      2.0416 

26.935    3-0884 

1765 

25.561      2.0417 

42.376 

3.0881 

1770 

35.770      2.0418 

57.816 

3.0879 

'775 
i  780 

45-979 
56.  189 

2  .  04  i  9 

2.  O42O 

21  54  '3-  255      3-0877 
28.693      3-0875 

1785 

20  34  6.399 

2.  042  I 

44-130      3-0872 

1790 

16.610 

2.  O422 

59-  565 

3.0870 

1  795 

26.822 

2.0425 

21  55  15.000 

3  .  0868 

1800 

37-054 

2.0424 

30-433 

3.0866 

1805 

47.246 

2.0425 

45-865 

3-0863 

18  10 

57-459 

2.  0426 

2  I  56   1  .  296 

5.  0861 

1815 

20  55  7-675 

2.  0428 

16.726 

3-0859 

1820 

17-887 

2.0429 

32.155 

3-0857 

1825 

28.  101 

2.0450 

47-  583 

3-0855 

1830 

38.316 

2.0451 

21  57  5.010 

3-0852 

i835 

48.532 

2.0452 

i8.435 

5.0850 

1840 

58.748 

2-0433 

33-86o 

3.0848 

1845 

20  56   8.965 

2-0434 

49-  285 

3  .  0846 

1850 

19.  l82 

2.0435 

21  58  4.706 

3.0844 

1855 

29.599 

2.0456 

20.  127 

5.0842 

1860 

59.618 

2-0457 

35-548 

5  .  0840 

1865 

49.856 

2.0458 

50-967 

5.0838 

1870 

20  57  0.056 

2.0459 

2'  59  6.585 

3-0835 

i8/5 

10.  276 

2.0440 

2  I  .  8O2 

3-0833 

1880 

20.  496 

2.0441 

37-218 

3-0831 

1885 

30-7'7 

2-0445 

52.633 

3.0829 

1890 

40.  959 

2.0444. 

22   O   8  .  048 

3.0827 

1895 

51.  161 

2.0445 

25.461 

3-0825 

1900 

20  58   ,.585 

2.0446 

38.873 

3-0823 

68 


RIGHT  ASCENSIONS  OF   FUNDAMENTAL   STARS. 

Mean  right  ascensions,  <£c. — Continued. 


Year. 

a  PlSCIUM  AUSTEALIS. 

a  PEGASI. 

R   A. 

Ann.  Var. 

R.  A. 

Ann.  ATar. 

ll.     'III.           S. 

s. 

It.     111.           S. 

s. 

1/5° 

22  43  46.  710 

3-3551 

22     52     19.531 

2-9775 

J755 

22  44     3.483 

3-3540 

34-4I9 

2-9778 

i  760 

20.  250 

3-3529 

49-  309 

2.9780 

1765 

37.012 

3-r35'8 

22   53     4-  199 

2.9783 

'7/o 

53-768 

3-3507 

19.  091 

2.9785 

i/75 

22  45    10.  518 

3-3496 

33-985 

2.9788 

1780                        27.  264 

3-3485 

48.879 

2.9790 

1785                        44-003 

3-3474 

22   54     3-  775 

2-9793 

1790 

22  46     o.  737 

3-3463 

18.  672 

2-9795 

J795 

17.466 

3-3452 

33-570 

2.9798 

1  800 

34-  189 

3-3441 

48.470 

2.  9801 

1805 

50.907 

•3-3430 

22   55     3-37' 

2.9803 

1810 

22  47     7.619 

3-34'9 

18.273 

2.  9806 

1815 

24.326 

3-3408 

33-  177 

2.  9809 

1820 

41  .  028 

3-3397 

48.082 

2.  981  I 

1825 

57-  724 

3-3387 

22     56        2.988 

2.9814 

1830 

22    48     14-4H 

3-3376 

17.  896 

2.9817 

1835 

31.  ioo 

"*       •*»  i  (~\  ^ 

o-  oo°5 

32.805 

2.  9819 

1840 

47-779 

3-  3354 

47-715 

2.  9822 

1845 

22  49     4.454 

3  •  3343 

22   57     2.627 

2.9825 

1850 

21.   123 

i       f\  *>  -7  *5 
O-  J  JOO 

17.540 

2.9828 

1855 

37.786 

3-3322 

32.455 

2.9830 

1860 

54-445 

3-33'1 

47-370 

2.9833 

1865 

22     50     II.  097 

3-3300 

•22     58        2.288 

2.9836 

1870 

27-745 

3.3290 

I  7.  2O6 

2.9839 

i875 

44-387 

3-3279 

32.  126 

2.9841 

1880 

22     51         I.O24 

3-3268 

47.048 

2.9844 

1885 

17.656 

3-3258 

22   59      1.971 

2.9847 

1890 

34.282 

3-3247 

16.895 

2.9850 

1895 

50-  903 

3-3236 

31.821 

2.9853 

1900 

22     52        7.518 

T,  .  T.226 

\.J         ^J 

46.748 

2.9856 

CORRECTIONS  FOR  ORBITAL  MOTION  OF  SIRIUS  AND  PROCYOX. 

Periodic  term*  to  be  <ij>]>//crl  tu  tin'  riyltt  d-yci'iisionti  of  Sir  ins  and  F  roc  yon. 


69 


a  CAMS  MAJOKIS. 

Year. 

Year. 

Year.                                   P 

s. 

1750.6 

1800.  o 

1849.4                       4-  .026 

1751.6 

i  80  i  .  o 

—  20 
1850.4                       +.006 

1752.6                     1802.0 

1851.4                        -.014 

1753.6                     1803.0 

1852.4                        --031       ^ 

i  754.  6                        1804.  o 

i853-4                        --047 

1755.6                        1805.0 

1854.4                        -.062 

i  756.  6                        1806.  o 

1855.4                     -.076_ 

1757.6                        1807.0 

1856.4                     -.088 

,758.6 

i  808.0 

1857.4                     -.099 

1/59-6 

i  809.  o 

—  I  o 

1858.  4                       -  .  109 

,  760.  6 

1810.  o 

1859.4                        -.n8_ 

1761.6 

i  8  1  i  .  o 

1860.4                       —.126 
—    6 

1762.6                        1812.0 

,861.4                          -.132 

1763.6                        1813.0 

,862.4 

-.138 

i  764.  6                        1814.0 

1863.4                        --'43 

1765.6                        1815.0 

1864.  4 

-.147" 

i  766.  6                        1816.0 

1865.4 

-.149" 

i  767.  6                        1817.0 

1866.4 

-  -'5' 

1768.6                       1818.0 

1867.4 

~  •  '  5- 

i  769.  6                        1819.0 

1868.4 

o 
—  •'5-   , 

-4-     i 

1770.6 

1820.  o 

1869.  4 

-•'5' 

-4-         T 

1771.  6 

1821  .  o 

1870.4 

-•'49  , 

1772.6 

1822.  o 

[871.4 

-.i464 

"773-6 

1823.  o 

,872.4 

—  .142 

4-     c 

1774.6                      1824.0 

'873-4 

••'•>'+    6 

'775-6 

1825.  o 

,874.4 

-•'3i 

1776.6 

1826.0 

1875-4 

+    7 

*      1           — 

1777.6 

1827.0 

1876.4 

-•"'4 

-r-     O 

,778.6 

1828.0 

.877.4 

-.108^ 
-t-  10 

1779.6 

1829.  o 

1878.4 

-  .098  ] 

-r-  I  2 

1780.  6 

1830.0 

1879.4 

-.086] 

i     '  2 

i  78  1  .  6 

I  83  I  .  0 

1880.4 

~~'°74+  I" 

i 

7o 


RIGHT   ASCENSIONS  OF   FUNDAMENTAL   STARS, 

Periodic  tcnns,  d'c. — Continued. 


a  CANIS  MAJOKIS  —  Continued. 

Year. 

Year. 

Year. 

P 

s. 

1782.6 

1832.0 

1881.4 

—  .061 
+  '5 

1783.6 

1833-0 

1882.4 

1784.6 

1834.0 

1883.4 

—  -°3°  +  ]8 

1785.6 

1835-0 

1884.4 

—  .OI  2 

+  19 

.1786.6 

1836.0 

1885.4 

0    t 

+  .007 

+  20 

1787.6 

1837.0 

1886.4 

+  -°27+22 

1788.6 

1838.0 

1887.4 

+  -049    , 

1789.6 

1839.  o 

1888.4 

_1_          "7                ~° 

i  790.  6 

1840.  o 

1889.4 

+  -°96+24 

1791.6 

1841  .0 

1890.  4 

+  'I2°    1     oi 

1792.6 

1842.  o 

1891  .  4 

+  .141 

1793.6 

1843.0 

1892.4 

+  -i52_ 

1794.6 

1844.  o 

1893.4 

+  -I47 

1/95-6 

1845.0 

1894.4 

+  -130 

i  796.  6 

1846.0 

1895.4 

+  -'°7 

1797.6 

1847.0 

1896.  4 

+  .082 

1798.6 

1848.0 

1897.4 

+  -osS 

1               >-'                     0-1 

1799.6 

1849.0 

1898.4 

J_                 ~° 

1800.  6 

1850.0 

1899.4 

+  -OI4 

—  20 

i  80  i  .  6 

1851.0 

1900.  4 

—  .006 
—  18 

1802.6 

1852.0 

1901.4 

—  .024 

CORRECTIONS  FOR  ORBITAL  MOTION  OF  SIRIL'S  AND  PROCYON. 

Periodic  terms,  ((•<: — Continued. 


a  CANIS  MINORIS. 

i 

Year. 

Year. 

Year. 

Year. 

p 

s. 

1750.0 

i  790.  o 

1830.0 

1870.  o 

—  -°45 
—    8 

1751.0 

1791.0 

1831.0               1871.0 

-  -°53 

1752.0 

1792.0 

1832.0              1872.0 

-.060 

—      t; 

1  753-o 

'793-0              1833.0              1873.0 

—  .065 

1754-0 

1794.0              1834.0              1874.0 

—  .068  ~ 

'755-0 

1  795-o              1835.0              1875.0 

—  .O7O 

1756.0 

1796.0              1836.0              1876.0 

O 

—  .070 

'757-o 

1797.0 

1837.0              1877.0 

-  .068  4 

_[._            ^> 

1758.0              1798.0 

1838.0              1878.0 

—  .065            ° 

i759-o              1799-0              1839.0              1879.0 

-  .060  + 

1760.0              1  800.0              1840.0              1  880.0 

--053^     I 

i  761  .  o 

1  80  1.  o 

1841  .  o 

1881.0 

~  -045   , 

1762.0 

1802.0              1842.0 

1882.0 

—  .o;6 

-4-   TO 

1763.0 

1803.  o 

1843.0 

1883.0 

.,      TT    J  U 
—  .026 

1764.0 

i  804  .  o 

i  844  .  o 

1884.0 

/-+    I0 
—  .Ol6 

1765.0 

1805.  o 

1845.0 

1885.0 

—  .006 

1766.  o 

i  806  .  o 

1846.  o 

1886.0 

+    I   1 

1767.0 

1807.0 

1847.0 

1887.0 

+  .oist10 

1768.0 

1  808.0 

1848.0 

1888.0                         +  .026~f 

1769.0 

1809.  ° 

1849.  o 

1889.0 

.             ,  +  10 
+  •036 

1770.0 

1810.  o 

1850.0 

1890.  o 

+  -045     , 

1771.0 

181  1  .  o 

1851.0 

1891  .  o 

+  -053  j 

1772.0 

I  8  I  2  .  O 

1852.0 

1892.  o 

+  .o6o4 

'773-0 

1813.0 

1853-0 

1893.0 

+  -065 

1774.0 

1814.0 

1854.0 

1894.  o 

_j_      2 

'775-o 

1815.0 

1855-0 

1895.0 

+  -O7° 

1776.0 

1816.0 

1856.0 

1896.  o 

O 
+  -070 

1777.0 

1817.0 

1857.0 

1897.0 

+  .068 

1778.0 

1818.  o 

1858.0 

1898.0 

+  .o65_    * 

1779.0 

1819.0 

1859.0 

1899.  o 

+  .060 

1780.0 

1820.  o 

1860.  o 

1900.  o 

+  -054_ 

1781.0 

1821.0 

i  86  i  .  o 

i  90  i  .  o 

+  .046 

—    9 

1783.0 

i 784. o 
1785.0 
1786.0 
1787.0 
1788.0 
i 789. o 
i 790. o 


RIGHT   ASCENSIONS   OF   FUNDAMENTAL   STARS. 

Periodic  twins,  (C'c. — Continued. 


Your. 

Yea 

1782.  o 

1822 

a  CANIS  MIXORIS — Continued. 


1823.0 
1824. o 
1825.0 
1826.0 
1827. o 

1828.0   .  i 

1829. o 
1830.0 


Year. 


1862.0 
1863.0 
1864. o 
1865.0 
1866. o 

1867.0 
1868.0 

1869. o 
1870. o 


1902. o 

1903.0 

1904. o 

1905.0 

1906. o 
1907.  o 
1908. o 
1909. o 
1910. o 


s. 

+  •037 
+  .027 
+  .017 

-f-  .006 

—  -005 

—  .015 

—  .027 

-•037 

—  .046 


—  IO 

—  10 
-  1 1 

—  1 1 


IO 

9 


HILL'S  FORMULA  FOR  SECULAR  VARIATION  OF  ANNUAL  MOTION  OF  STARS.  73 

HILL'S  FORMULA  FOR  THE  SECULAR  VARIATION  OF  THE  ANNUAL 

MOTION  OF  THE  STARS. 

[From  Star-Tables  of  the  American  Ephcmeris,  p.  xix.] 

These  formulae,  being  more  complete  than  those  usually  employed,  are  given  here, 
for  convenience  of  reference,  in  a  slightly  modified  form. 
Put- 

ju,   the  proper  motion  in  right  ascension,  expressed  in  seconds  of  time; 
//,  the  same  in  declination,  expressed  in  seconds  of  arc  ; 
p,  p',  the  annual  precessions,  expressed  in  the  same  Avay  with  /*  and  //', 
respectively. 

We  then  have  — 

p  rr  m  -\-  n  sin  a  tan  8 
p'  —  n  cos  a 
da 

^  =  'l  +  " 
(16 
~di  =  P+f* 

10°f  TT  =  [7-9878]  n  cos  a.  tan  8  -f-  [6.81  17]  n'  sin  a  sec2  S 

d  u' 
loo-'™   zz  —  [9.  1640]  JLI  sin  a 


ioo         =  +  .  00322 

-  [6.6338]  j> 

+  [7-9878]  (p  +  2  /()  cos  a  tan  '5 
+  [6.8117]  (y  +  2  /O  sin 
-f  [6.9866]  n  //'tan  «5 
fl*fi  on     / 

too  -/r,  -  =  -[6.6338]  p' 

—  [9.1640]  (^  +  2  /<)  sin  a 
-[8.7367]  >"a*in  2  r5 


Struve's  values  of  m  rt«<7  n. 
Year.  m  n  log  «  »  log  « 


s. 

s. 

1750 

3.06987 

I  -33767 

0.12635 

20.0650 

1.30244 

1800 

3.07082 

1-33738 

o.  i  2626 

20.0607 

I  -30235 

1850 

3.07177 

I  -33710 

o.  12616 

20.0564 

1.30225 

1900 

3.07272 

1.33681 

o  .  i  2607 

20.0521 

I  .30216 

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